Trump, who has riled establishment Republicans, easily defeated his two rivals for the Republican presidential nomination, U.S. Senator Ted Cruz of Texas and Ohio Governor John Kasich, U.S. television networks projected.
On the Democratic side, Clinton stretched her advantage in the Democratic contest by winning Arizona, routing U.S. Senator Bernie Sanders of Vermont.
The contests in Arizona and Utah took place on the same day that attacks in Brussels killed at least 30 people, adding to Americans' concerns about the threat from Islamist militants.
"This is a time for America to lead, not cower," Clinton told supporters in Seattle in a victory speech she used to attack Trump and Cruz for views she said were out of step with most Americans.
Sanders was unbowed in defeat, saying national polls showed him gaining on former secretary of state Clinton.
"When we began this campaign about 10 months ago we were three percent in the polls, about 70 points behind secretary Clinton. As of today the last poll that I saw we are five points behind and we are gaining," he said.
Long lines of voters were reported in both states.
Trump, the New York billionaire and former reality TV star, has ridden an anti-Washington message to become the favorite for the nomination. This has left a flagging anti-Trump effort with faint hopes of stopping him at the Republican national convention in July.
"Much bigger win than anticipated in Arizona. Thank you, I will never forget!" Trump said on Twitter. "Hopefully the Republican Party can come together and have a big WIN in November, paving the way for many great Supreme Court Justices!"
Trump's hardline immigration message is popular in Arizona, one of the U.S. states that borders Mexico, and he leads in polls. He lags behind top rival Cruz in Utah.
Arizona will award its entire slate of 58 delegates to Trump.
In Utah, the state's 40 delegates will be awarded proportionate to the popular vote, unless a single candidate captures at least 50 percent of the vote, in which case that person will be awarded all the delegates.
On Monday, Trump warned against efforts to deny him the nomination if he falls short of securing the 1,237 delegates needed ahead of the July convention. Trump now has 678 delegates.
"I think it is going to be very hard for them to do," Trump said on CNN of any effort to deny him the nomination if he falls short. "I have millions of votes more than anybody."
Clinton showed her strength yet again in Arizona and looked to have a solid path to the Democratic presidential nomination. The two nominees from each party will meet in the Nov. 8 presidential election.
Sanders is looking for wins in many of the six Democratic contests this week. Alaska, Hawaii and Washington will vote on Saturday. Clinton will keep adding to her delegate total even if she is not the winner in a given state because Democratic delegates are awarded proportionally in all states.
Tuesday's Republican contests were the first since U.S. Senator Marco Rubio of Florida dropped out a week ago after Trump drubbed him in his home state.
Kasich is the only other candidate still in the race, splitting the anti-Trump vote with Cruz.
In Arizona, Trump had the backing of former Republican Governor Jan Brewer and Maricopa County Sheriff Joe Arpaio, two of the most prominent supporters of a crackdown on illegal immigrants.