With 42 percent of precincts reporting, Mr. Rangel was declared the winner of the Democratic primary with 52.8 percent of the vote. His closest competitor, state Sen. Adriano Espaillat, was trailing with 31.1 percent, while a third candidate, former Democratic National Committee official Clyde Williams Jr., held 11.7 percent.
In Utah, Mr. Hatch appeared to be coasting to victory in the Republican primary. With 28 percent of precincts reporting, Mr. Hatch led with 68 percent of the vote to 32 percent for former state Sen. Dan Liljenquist.
Utah was also the last state to hold its Republican presidential primary, a contest that likely nominee Mitt Romney, a Mormon and favorite of Utah voters, was expected to win handily.
Mr. Romney had 92 percent of the vote with 28 percent of precincts reporting.
In another closely watched race, Colorado Rep. Doug Lamborn held a strong lead over businessman Robert Blaha in the Republican primary.
Early returns showed Mr. Lamborn ahead by a margin of 61 to 39 percentage points.
One unexpected glitch: A fast-moving wildfire forced the closure of El Paso County’s main ballot-counting center, meaning that final results were not expected until Wednesday.
The winner of the Republican Senate primary in staunchly conservative Utah is virtually guaranteed to win the November election, as is the victor in the New York Democratic congressional primary, given the districts’ lopsided voter registration.
Both Mr. Hatch and Mr. Rangel are venerable Washington fixtures with nearly eight decades of service between them. Mr. Hatch, 78, is seeking his seventh term in the Senate, while a win for Mr. Rangel, 82, would give him a 22nd term in the House.
Of the two, Mr. Rangel faces steeper odds. He was censured on the House floor 18 months ago for income-tax violations and removed as chairman of the House Ways and Means Committee. His redrawn Harlem-based district now has a Hispanic majority, which is expected to benefit Mr. Espaillat, who would become the first American of Dominican descent to serve in Congress.
Mr. Rangel won the endorsement of Democratic Gov. Andrew Cuomo, while Mr. Williams, a former domestic policy adviser to President Clinton, was endorsed by the New York Times and the New York Daily News.
In Utah, Mr. Hatch faced what has become an occupational hazard for Republicans of a certain age: a younger, more conservative challenger backed by the tea party. Former state Sen. Dan Liljenquist ran an aggressive campaign aimed at ousting Mr. Hatch just as Mike Lee upended then-Sen. Robert F. Bennett in 2010.
Mr. Hatch had studied the 2010 playbook, and he launched a vigorous re-election effort aimed at avoiding the mistakes of the Bennett campaign. Mr. Hatch wooed tea party groups, courted delegates to the Utah Republican Convention, and nearly won the nomination outright in April with 59.2 percent of the delegate vote.
Mr. Liljenquist tried to paint the veteran senator as hopelessly out of touch with the party’s shift to the right on federal spending and budget reduction. He hammered Mr. Hatch for voting for spending programs and earmarks without designating how they would be funded, and called him part of the reason for the nearly $16 trillion federal debt.