Harold Solomon defeated a sometimes troublesome opponent, Bernie Mitton, 7-5, 6-4, 6-2, yesterday to clinch victory for the United States in its Davis Cup American Zone North Section final against South Africa.

Solomon's two-hour, 20 minute triumph, after he luckly escaped four set points against him as he served at 5-6 in the first set, gave the U.S. a 3-1 lead in the controversial best-of-five match tennis series.

Vitas Gerulaitis, who gave the U.S. its first point with a four-set victory over Mitton in the opener, was to play Byron Bertram in the final singles.

Solomon, 25, predicted that the U.S., which has won the premier team prize in tennis 24 times since it was introduced by Dwight F. Davis in 1900, would go on to win three more series and capture the sterling silver punchbowl for the first time since 1972.

But he conceded that the U.S. may have a difficult time in its next contest, the American Zone final against South Section winner Chile at Santiago Sept. 15-17.

That is the week after the U.S. Open, which will be played on hard courts for the first time. Some top American players already have conflicting commitments for those dates, and most will be tired.

In any event, there will be little time to acclimate a team to the slow clay surface and conditions at Santiago's Estadio Nacional, where Jaime Fillol, Hans Gildemeister and Patricio Cornejo teamed to defeat Argentina this weekend.

Solomon, who now has a 6-3 record in Davis Cup zone matches since 1972, said he expected to be available to play in Chile. The U.S. has not won the American Zone since 1973, having been humbled by Colombia in 1974, Mexico in 1975-76, and Argentina last year.

Solomon had secured the decisive point in a Davis Cup series once before, beating Raul Ramirez in straight sets in the fourth match of a 4-1 victory over Mexico in 1973.

Only about 50 protestors demonstrated outside Vanderbilt University's Memorial Gymnasium yesterday against the apartheid policy of the South African government.

Anti-South African demonstrations here peaked Saturday, when some 2,000 protestors marched three miles from the Tennessee state capitol to the gym and joined hundreds of others for an antiapartheid rally.

Officials of the National Association for the Advancement of Colored People, which organized the march, termed it a success in its intent to bring to the attention of the American people conditions in South Africa and the racial separation policy of its government.

Heavy security precautions remained in effect yesterday as 150 policemen, three for each protestor, remained on duty around Memorial Gym.

There was one minor incident during the first set of the Solomon-Mitton match. A middle-aged man wandered from the press section past uniformed police deployed at the end of each aisle and onto the court. He held up a poster showing a clenched fist covered with blood.

He mumbled, "These tennis matches are disgusting. You all have blood on your hands," but his words were barely audible, even to spectators in the first rows of the gymnasium. Two policemen quickly grabbed him and ushered him off.

The man was later identified as Gerald Hornsby, a copy editor for The Tennessean, a Nashville daily newspaper. He apparently had obtained press credentials from someone else on his newspaper's staff. He was not arrested.

Only 1,260 spectators attended yesterday's matches, bringing the three-day total to 4,374. The local organizers had said they needed three-day attendance of 12,000 to break even financially. Profits were to go to the Vanderbilt tennis program.

The U.S. Tennis Association released the university from its $50,000 guarantee, however, and Joe Carrico, chairman of the USTA Davis Cup Committee, said the association would split the expected modest deficit with Joe Davis, a wealthy local businessman and Vanderbilt board member who had agreed to underwrite the university's financial risk in staging the matches.

Carrico blamed the low attendance on publicity given the demonstrations and fears of violence.

This likely was South Africa's last appearance in the Davis Cup in the forseeable future. The USTA is committed to working for its removal from international tennis competition, either by voluntary withdrawal or by expulsion, at the annual meetings of the Davis Cup Nations and International Tennis Federation in July. The move is expected to succeed.

"If South Africa is going to be kicked out, there's not much you can do about it. But up to now we haven't thought about it this way, as our last Davis Cup match," said Mitton.

Solomon, who had a 2-2 record against Mitton the last three years, normally finds him a much tougher opponent than Bertram, whom he beat in four sets Friday night.

"Mitton moves a little better. They can both hit a lot of big shots off the forehand, but Mitton is in a little better shape, he's faster, and he can move you around a little bit more," Solomon said.

"He does more with his backhand. He can chip it down the line or cross-court and make me run. He's more dangerous for me because he can hit himself into a streak and hit a lot of winners."

But Solomon kept battering his hard, deep ground strokes at that backhand until it crumbled, and won a number of points when Mitton over-anxiously went for big forehand and overhit them.

Mitton broke for 2-0 ad 4-2 lead in the first set, but each time Solomon broke right back in the next game.

Milton thought he had held for 3-0 with an ace, but his serve was called a fault, and he looped a backhand wide after Solomon drilled a backhand down-the-line winner to get to his third break point. Solomon broke at love to get back to 3-4.

Mitton. 24, had his biggest chance in the 10th game as Solomon struggled on his serve, double-faulting three times in that game before holding for 5-5 after seven deuces and four set points.

The gritty 5-foot-6, 138-pounder brushed off the set points with a backhand down-the-line pass, a hard forehand approach that surprised Milton and forced a backhand error, and two bad errors by Milton off rallies - first a hestitant backhand, then a miss-hit forehand.

Solomon broke at 15 in the next game, again inteimidating Milton into sailing a backhand by abandoning his usual backcourt game and coming in behind a forehand approach. Solomon then served out the set after three deuces, saving one break point as a short, weak backhand clipped the net and cribbed over for a winner.

Thereafter Milton became increasingly discourage and ragged.