Governments in both the West and the East today welcomed the victory of West German Chancellor Helmut Schmidt's left-center coalition in elections yesterday -- although for somewhat different reasons.

At the same time, political observers here weighed the pluses and minuses for Schmidt in having to govern now with a stronger junior centrist coalition partner.

Final election results showed that Schmidt's government more than quadrupled its formerly narrow majority in the lower house of parliament, gaining 18 seats for a new margin of 45 seats over the conservative oppositon. The coalition of Social Democrats and Free Democrats took a combined 53.5 percent of the vote over the Christian Democrat-Christian Social Union combination, which won 44.5 percent.

President Carter phoned congratulations today both for Schmidt, leader of the Social Democrats, and Foreign Minister Hans-Dietrich Genscher, leader of the Free Democrats -- the party most responsible for the government's gains.

While U.S. officials expected Schmidt to win, they were surprised by the strong showing of the Bonn government's liberal junior partner. The strengthening of West Germany's political middle seemed to have a reassuring effect on Washington after misunderstandings and a sense of mistrust over Bonn's detente policies following the Soviet invasion of Afghanistan.

Moscow was also pleased by the election outcome, but more because of Schmidt's reelection than Genscher's triumph. The Soviets, who also had reckoned with another four-year term for the West German chancellor, saw in the result a reinforcement of European detente policy over the campaign charges of the opposition that Bonn was drifting under Soviet influence

"The victory of the Social Democratic and Free Democratic parties . . . shows very clearly that the greater part of the West Germany population favors a continuation of the policy of detente and international cooperation," the Soviet news agency Tass reported.

West Europeans added their approval as well, viewing the vote's outcome as a boost for Europe's own growing political assertiveness, a trend embodied by the commanding figure of Schmidt, 61, in both his style and policies.

First on the phone last night to congratulate the chancellor, according to Schmidt, was French President Valery Giscard d'Estaing -- a fact that reflects not only the personal friendship between the two leaders but, more significantly, the cornerstone relationship to an increased world political role for Western Europe. While Giscard has promoted this idea publicly more than Schmidt, the French leader counts on his partner in Bonn to fashion East-West, Middle East and economic policies more independently of the United States.

A victory by the Christian Democrats arguably would not have changed the broad outline of Bonn's foreign policy, more or less fixed at the moment by recent history and geography, but it could well have changed the accents, giving it a more pro-U.S. tone, which in turn would have made it less appealing for the French.

Western Europe's smaller countries, meanwhile, appeared happy for Genscher's success. It was hoped that his party might restrain Schmidt from overly fortifying the Bonn-Paris axis, which sometimes worries the less powerful nations of the European Community.

Schmidt's reelection generally signaled to the world the continuation of a sure-handed, if controversial, Bonn foreign policy at a moment when the Western allies are facing critical questions of human rights and disarmament at talks with the East Europeans soon to begin in Madrid and Geneva.

Additonally, Schmidt has placed two important themes on the table of the European Community -- reforming the European agricultural system and restructuring the community's budget.

The chancellor may have to water down his ideas for agricultural reform as a result of the enlarged representation in the Bonn government of the Free Democrats -- a subject on which Agricultural Minister Josef Ertl, a Free Democrat, is known to have misgivings.

Overall, Schmidt's governing position has been made both more easy and more complicated by the election. The need to accomodate a more influential liberal party should strengthen the chancellor's hand against the left wing of his own party, which is opposed to NATO nuclear rearmament and to the development of atomic power domestically.

It will also mean tougher going for Schmidt in areas where the coalition partners have differed in the past -- on labor policy, pension reform and the national budget.

In foreign policy, the differences between Genscher and Schmidt have tended to be more noticeable in nuances than substance, with the foreign minister being a more outspoken supporter of the United States than the sometimes restrained chancellor. More friction between the two comes on domestic issues.

Despite the Free Democrats' advances -- they won 14 more seats in the 497-seat parliament for a total of 53 -- they have decided not to press for a change in the balance of power in the new Cabinet. At present, the liberals fill four key Cabinet posts -- foreign, economic, interior and agriculture. Party sources said, however, that the Free Democrats may capitalize on the election result by seeking a couple of extra appointments at the deputy minister level.

Genschr, 53, known as a decent and skilled politician, is expected to exercise his party's greater influence on Schmidt sparingly and quietly. However, should his low-key tactics within the coalition run into too much contradiction from the Social Democrats, political observers figure he may now be inclined to press his aims more demonstrably.

"The question is whether Genscher takes advantage and gives the Socialist-liberal coalition a new thrust, such as it had at the beginning of the 1970s, when it was important to carry through the detente policy, the thrust that it needs again to meet the challenges of the 1980s," the daily Stuttgarter Zeitung commented today.

The paper suggested that Genscher's telling choice would be between this progressive strategy and a strategy simply of political maneuvering between bonn's two larger political parties.

Meanwhile, the conservative Christian Democrats, whose loss of 4.1 percentage points gave them their worst result since the first parliamentary election in 1949, sought to avoid party sniping today against their chancellor candidate, Franz Josef Strauss, leader of the Christian Social Union in Bavaria. Christian Democratic leader Helmut Kohl, declaring that his party would retain its tie with its sister party, told reporters here that there was no cause "to look back and start to debate to apportion blame." The conservatives still hold the largest bloc in the lower house with 226 seats and have a legislation-blocking majority in the upper house.