CONCORD, N.H.--Vice President George Bush sped past Sen. Bob Dole (R.-Kan.) in New Hampshire's Republican presidential primary yesterday, while Massachusetts Gov. Michael S. Dukakis won the Democratic contest handily.

Bush rebounded from a third-place finish in last week's Iowa caucuses, winning 38 percent of the GOP vote to Dole's 29 percent.

Rep. Jack Kemp (R.-N.Y.) narrowly defeated Pete du Pont and Pat Robertson in the competition for third place, emerging as a conservative alternative to the front-runners. But the group was clumped far behind Bush and Dole.

Dukakis won 36 percent of the vote and said his showing would "give us a very, very strong boost" going into the delegate-rich Southern primaries ahead. The distant but hotly contested second place went to Rep. Richard Gephardt (D.-Mo.), 20 percent, with Sen. Paul Simon (D.-III.) at 17 percent.

"I love New Hampshire," the Missouri congressman said. But Simon, a narrow second-place to Gephardt in Iowa, conceded nothing.

A triumphant Dukakis told his supporters, "Ten months ago we launched a campaign for the future of America, a campaign for good jobs and real opportunity for every citizen...a campaign to get our fiscal house in order."

In a hint that his southern campaign will concentrate on urban areas, Dukakis mentioned Miami, Atlanta, Norfolk and Selma in his speech. The governor insists, however, that he will campaign in all parts of the South.

The speech proposed unity and optimism, avoiding specifics and political details.

Boston-based political consultant Michael Goldman said the governor's victory left him strongly positioned for a run in the South. He said Gephardt is nearly out of money and Simon is all but out of the race. Goldman said the crowded field actually helped the governor by preventing any one candidate from garnering an anti-Dukakis vote.

Goldman said Dukakis plans to spend nearly $4 million in the South, more even than Sen. Albert Gore (D.-Ten.), who has planned a $1 million media campaign.

The effort earned Dukakis a bronze medal in Iowa, he said, and in next-door New Hamshire, "We went for the gold and we won it."

CBS, reporting on the basis of interviews with voters leaving their polling places, said President Reagan's popularity in the Granite State helped Bush considerably. Other pollsters concurred, and said Dukakis also profited from his long tenure as governor of a neighboring state.

With 94 percent of the precincts reporting, Dukakis was likely to receive 9 Democratic convention delegates, Gephardt 6 and Simon 3. That would raise the national totals to 44.50 for Dukakis, 39 for Gephardt, and 33 for Simon. In all, 2082 delegates are needed to win the Democratic presidential nomination.

Jesse Jackson and Sen. Albert Gore Jr. (D.-Tenn.), were denied the boost they sought as the primary campaign moves to their native South, trailing with 8 percent and 7 percent of the vote. Former Arizona Gov. Bruce Babbitt had 5 percent and was a candidate for withdrawal. Long-ago front-runner Gary Hart had 4 percent.

Gore already had commitments for 10.55 delegate votes at the convention, Jackson had 9.80 and Babbitt four.

In the Republican race, Bush led with 11 New Hampshire delegates, Dole seven, Kemp three and du Pont two. Those results would raise their national totals to 61 for Bush, 42 for Dole, 35 for Kemp and two for du Pont. In all, 1139 delegates are needed to win the Republican presidential nomination. Empty-handed yesterday, Robertson already had won eight delegates.

The final number of delegates was not available last night, but the latest unofficial tally of the popular vote, with 99 percent of the precincts reporting, shows the following:

Dukakis, with 41,185 votes, received 37 percent. Gephardt, remaining at 20 percent, received 22,464 votes. Simon's 17 percent was also stable, with 19,432 votes. Jackson, at 8 percent, received 9062; Gore, 7 percent or 7698; Babbitt, 5 percent or 5334; Hart, 4549 or 4 percent, and others with 2601 votes or 2 percent.

The comparable results for the Republicans showed Bush with 54,915 or 38 percent; Dole, 41,746 or 29 percent; Kemp, 18,367 or 13 percent; du Pont, 14,448 or 10 percent; Robertson, 13,839 or 10 percent; Haig, 522, or less than 1 percent. The remaining Republican candidates received a total of 511 votes.