The Republican presidential candidate Mitt Romney has put his faltering campaign back on track, holding off a strong challenge from rival Rick Santorum in a tight primary race in Michigan and coasting to victory in Arizona.

Speaking at an election party in Novi, on the outskirts of Detroit, Romney described it as a "decisive moment". He admitted he had not won by a lot but said it was enough. "Great victory in Arizona. Thank you Michigan. What a win. This is a big night. A week ago the pundits and the pollsters were ready to count us out," he said.

He pointedly did not mention Santorum, who fought a tough campaign that on Monday led to Romney to accuse him of dirty tricks.

Defeat in Michigan would have been near-catastrophic for Romney. It is his native state and his father was a popular three-term governor here. The victories put a brake on Santorum, who had been building momentum after three unexpected wins earlier this month.

Santorum phoned Romney to concede before all the votes had been counted and then spoke in Grand Rapids, Michigan, to his own supporters. He did not congratulate Romney during his speech, suggesting bad blood between the two, and instead suggested he is in for the long run, boasting about the recent rise in his profile. "A month ago they did not know who we were. They know now," he said.

It was a rambling speech that ranged from a rallying cry for liberty from the British in revolutionary days to US debt.

The race now moves to Super Tuesday next week when 10 states will be in play. Paul Begala, a political adviser in the Clinton White House, told CNN: "Armageddon for Romney will be Super Tuesday."

With 91% of the votes in Michigan counted, Romney had 41%, Santorum 38%, Ron Paul 12% and Newt Gingrich 6.5%. In Arizona, with 73% of the votes counted, Romney had 47.5%, Santorum 26%, Gingrich 16% and Paul 8.5%.

Romney won all 29 delegates in Arizona to add to his tally: he needs 1,144 to win the Republican nomination. Michigan's 30 delegates are distributed on a proportional basis.

Although Romney won the popular vote in Michigan, Santorum could emerge with more of the delegates because much of Romney's support came from one county and Santorum's was more widespread.

If Santorum had won he would have thrown the Republican race into chaos. Instead, Romney consolidated his status as frontrunner to win the Republican nomination and face Barack Obama for the White House in November.

Santorum lost some ground last week with a poor debate performance and a series of contentious comments that may have alienated moderates, from criticising universities to saying President Kennedy's speech calling for absolute separation of church and state made him want to throw up.

As a desperate last ploy, Santorum authorised the use of robocalls to urge Democrats, who are entitled to vote under the state's quirky election laws, to vote against Romney. Exit polls did show that one in 10 Democrats voted.

Romney's two victories provide him with a cushion going into Super Tuesday. He will make a contest of it in Ohio, where he will begin campaigning on Wednesday, and polls put him ahead in Massachusetts, Vermont and Virginia.

Santorum is ahead in the polls in the most important of the Super Tuesday states, Ohio, and in Tennessee and Oklahoma. Santorum is heading for Tennessee on Wednesday.

Exit polls in Michigan confirmed that Republicans, as in contests over the last two months, are largely unenthusiastic about the choice available to them.

The win in Michigan comes in spite of a series of gaffes by Romney over the last week that had appeared to hurt him. He reinforced his image of being ultra-wealthy last Friday when he spoke casually of his wife owning two Cadillacs and on Sunday when he said he was only interested a little in Nascar racing but knew many of the team owners.

In Arizona, Romney's victory was never seriously in doubt, but a crowd of about 300 supporters at the Hyatt hotel in Phoenix still voiced their relief at the result. The election was called for Romney the minute the polls closed at 7pm, a sign of the gap to Santorum, his nearest rival in Arizona.

"We always knew he would win but it's still good to have it confirmed," said Mike Hadley, aged 30, at the Phoenix victory party. "Romney is genuine – you know that he's running because he believes he can change America for the better."

Exit polls conducted by CBS News put the economy firmly at the top of the list of priorities of Arizona voters, with 48% saying they were concerned about economic wellbeing. By contrast, 12% said they were most worried about immigration, despite the prominence of the issue following Romney's promise to promote "self-deportation" of illegal immigrants.