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WASHINGTON — Rick Santorum dealt Mitt Romney a setback Tuesday night, winning the presidential primary in Missouri as Republicans in three states voted on a day that could produce a shift in the momentum of the 2012 race.

The former Pennsylvania senator’s victory in the non-binding “beauty contest” gave him bragging rights on a day when Romney had once been favored to sweep.

Romney had managed to put back-to-back victories together with wins in Florida and Nevada. Now he will have to regroup, with few opportunities to show strength until a pair of primaries that are still three weeks away.

Santorum, whose last victory came more than a month ago in Iowa’s leadoff caucuses, had his best showing since that day. Returns from caucuses in Minnesota show Santorum on his way to victory and Colorado also showed him running well.

Rep. Ron Paul of Texas, the only remaining candidate without a victory, has been targeting caucus states, including Minnesota and Colorado.

Delegates in both states will be apportioned later, and Missouri’s primary has no formal connection to delegate selection.

Romney, the former Massachusetts governor who carried Minnesota and Colorado in 2008, scheduled a celebration with supporters in Denver. Sounding confident, he told Colorado voters earlier in the day that their state would have “a little something to say about who our nominee is going to be.”

Santorum chose to stage his election-night event in Missouri, the largest of the three states voting Tuesday and the second-most populous to vote this year, behind Florida. One of his arguments to Republican voters is his ability to carry electoral battlegrounds, and Missouri in the past has fit that mold.

In 2012, Colorado and Minnesota are more likely to be competitive in the general election, while Missouri has become reliably Republican.

In Colorado and Minnesota, voters were gathered at precinct caucuses, where a straw poll of attendees was conducted to reflect the presidential preferences.

Tuesday’s caucuses were conducted under the same process as those in Iowa, which took weeks to sort out. Romney initially was said to be victorious, but after state party officials checked ballots, Santorum was declared the winner by 34 votes out of more than 121,500 cast.

The Missouri primary provided Santorum with his clearest opportunity yet to go head-to-head with Romney as the conservative alternative, because former House Speaker Newt Gingrich failed to qualify for the statewide ballot. Gingrich and Santorum have competed to eliminate the other and become the choice of the party’s most ardent conservatives.

Turnout in Missouri was reported to be low, and the candidates have spent little time campaigning there because the election is a “beauty contest” with no bearing on delegates to the national convention. State law required the primary to take place, and in most parts of the state, the presidential contest was the only item on the ballot.

But because of its early timing, Republican National Committee rules would have penalized Missouri for using the primary to select delegates. As a result, the state party opted to pick them in a separate caucus process that starts next month.

A memo, sent out from the Boston headquarters as voting was under way in Missouri and soon to begin in the other states, argued that “there is no way for any nominee to win first place in every single contest” and “we expect our opponents to notch a few wins too.”

“Romney is the only candidate prepared to compete in simultaneous campaigns across the country,” Rich Beeson, the Romney campaign’s political director, said in the memo. As for the others, even “success in a few states will not mean collecting enough delegates to win the nomination.”

Gingrich, in a tacit acknowledgment he did not expect to do well Tuesday, spent the day campaigning in Ohio. The state is one of 11 that will vote March 6, known as Super Tuesday.