Mississippi, where American politics often used to find its lowest common denominator, is offering the country a touch of class this year in the race between Sen. Thad Cochran (R-Miss.) and his Democratic challenger, former governor William Winter.

Although it has not yet become the close contest that many observers initially expected, it has injected a note of quiet civility into the leading Senate contests, most of which have been marked by a strikingly negative tone.

"Sometimes you achieve results whether you win or lose," Winter said the other day in a comment that appeared to reflect more optimism about the direction of politics in Mississippi than it did about prospects for his election.

Winter, although beginning to chip away at Cochran's once-formidable lead, still trails by a substantial margin, hobbled by a slow start at campaigning and fund raising, by President Reagan's popularity in the state and by Cochran's skills at political tightrope-walking.

The Mississippi contest is important nationally because Democrats are eyeing Cochran's seat as one of the half-dozen that they must pick up if they are to regain control of the Senate in the Nov. 6 elections, which most observers regard as possible but not likely.

But the real story of Mississippi this year lies less in its horse-race aspects than it does in the low-key dignity with which the affable, earnest Cochran and the scholarly, serious Winter are fighting it out on a political high ground that has not always been easy to find in the state.

Not only have they disavowed the race-baiting that characterized Mississippi politics for decades but both candidates are campaigning openly and hard for the votes of black as well as white Mississippians. Winter is heavily favored among blacks and Cochran's edge comes largely from the support of white conservatives.

In another departure from Mississippi politics of the past, the moderately conservative Cochran and the moderately liberal Winter are comfortable in the mainstream of their own national parties even though neither has aligned himself too closely with his party's presidential standard-bearer.

So far at least, they also have avoided the heavily negative, mudslinging campaign advertising that has characterized races in other states this year.

Among close southern contests for the Senate, Mississippi and North Carolina are poles apart on the race issue, with their historic roles reversed.

Only a couple of decades ago, North Carolina was leading the way toward a racially progressive "New South," while Mississippi was defiantly pulling in the other direction. Now, as Sen. Jesse Helms (R-N.C.) seeks to court white votes by focusing on support among blacks for Democratic Gov. James B. Hunt, two of the most popular figures in modern Mississippi politics are trying to outdo each other at biracial coalition-building.

There are blows to other Mississippi stereotypes as well.

Winter, for instance, is campaigning intensively in support of a nuclear freeze in this presumptively hawkish state, convinced that arms control is a strong, silent issue among many Mississippi voters.

Cochran gives a civics-class kind of critique of the Senate's inability to conduct its business at other than a snail's pace, suggesting modification of filibuster rules under which states like Mississippi used to be able to lay claim to power beyond their numerical strength in the Senate.

Long a force for racial moderation in the state, Winter had a mixed record of political success before winning the governorship, which he left last January on a positive note after winning approval for the most sweeping package of education reforms in the state's history.

But he vacillated over entering the Senate race, first accepting the chancellorship of the University of Mississippi and then rejecting it in favor a bid for the Senate, which he was under heavy pressure to make from influential national Democrats.

Although Winter says his voter surveys show he wasn't hurt in the process, the episode enabled Cochran to get a jump on him in campaigning and fund raising and to position himself as the above-party candidate.

"I want to represent Mississippi, not the interests of a national political party," Cochran said in just about the closest thing to a low blow in the campaign.

Although Cochran has a conservative, staunchly pro-Reagan voting record in the Senate, his approach is amiable, nondoctrinaire and even liberal on issues such as food stamp funding, which is highly important in the nation's poorest state.

On civil rights, he supported extension of the Voting Rights Act and voted to make the birthday of the Rev. Martin Luther King Jr. a national holiday.

Cochran had substantial black support as a House member and in his first race for the Senate in 1978 and is campaigning actively in black neighborhoods this year, despite expectations that Winter could get 85 to 90 percent of the black vote, which is about 30 percent of the total in Mississippi.

A bid for the House by black state Rep. Robert Clark (D) in the Mississippi Delta 2nd District may help boost the black voter turnout to Winter's advantage, and Winter was helped when a black independent decided at the last minute against entering the Senate race. It was in such a three-way contest that Cochran won his Senate seat.

But, as Cochran readily concedes, his campaigning among blacks forces Winter to divert attention from white voters in order to nail down his black base.

Although a first-termer, Cochran emphasizes his accumulating seniority and its potential value to Mississippi, especially in light of the fact that Winter is 62 and he is 46, with six years behind him in the Senate.

Mississippi puts a premium on seniority, especially for members of the appropriations committees, who keep the money flowing home. Rep. Jamie L. Whitten (D-Miss.) heads the House Appropriations Committee, while Sen. John C. Stennis (D-Miss.) is ranking Democrat on the Senate Appropriations Committee. Cochran also serves on Senate Appropriations as chairman of its agriculture subcommittee, which is of more than passing interest to rural Mississippi. He ranks third on the Senate Agriculture Committee and, by several quirks of electoral fate this year, conceivably could emerge as its chairman.

While Cochran characterizes the campaign as one of "promises" versus "performance," Winter contends that Cochran's performance, largely in support of Reagan's program of tax and spending cuts, has been out of step with the needs of an impoverished state like Mississippi.

In a speech to a group of young lawyers here last week, Winter took them on a forced march through a forest of statistics leading to the conclusion that only a massive infusion of private capital and federal aid ("even though it's a dirty word in certain circles") could drag the state into the nation's economic mainstream.

It was a cerebral speech that touched on concerns ranging from the environment to a nuclear freeze but lacked a simple clear message -- a missing ingredient in his entire campaign as a whole, some well-wishers have complained.

"You put this together with his hesitation about making the race and Thad's ability to please just about everyone . . . and it's no wonder he's having trouble," one supporter said.

Cochran is on his way to spending a record $2 million for a Mississippi campaign, about double what Winter hopes to raise and spend. But Winter campaign aides contend that he will be competitive with Cochran on television in the campaign's final days, with an added boost from Stennis, who plans an unprecedented commercial on Winter's behalf.