The lawsuit, filed in state Superior Court in Maricopa County, said poll workers told some voters to press a button after a machine had detected an "overvote."
The campaign contended that the decision disregarded voters' choices in those races, saying new voting machines were used on Election Day on Tuesday. The lawsuit suggested those votes could prove "determinative" in the outcome of the race between Trump and Democrat Joe Biden, who was declared the winner on Saturday by major television networks.
The Maricopa County Elections Department and a spokesman for Biden did not immediately respond to requests for comment.
Biden current leads Trump by 0.65%, or just over 21,000 votes in Arizona.
The Trump lawsuit, whose plaintiffs include the Arizona state Republican Party and the Republican National Committee, cited declarations by some poll observers and two voters that claimed the problem led to rejected votes.
It seeks an order requiring a manual review of purportedly overvoted ballots that were cast in-person. It also asks a judge to bar the certification of the vote until the review is complete.
The lawsuit says the review may "yield up to thousands of additional votes for" Trump.
On Saturday, a group of voters dropped a lawsuit that had made unsubstantiated claims alleging some votes cast for Trump were not counted in Maricopa County because voters used Sharpie pens.The county - which includes Arizona’s biggest city, Phoenix - last year rolled out new tabulation equipment that made Sharpie pens the best option on Election Day because they have the fastest drying ink, Megan Gilbertson, the Maricopa County Elections Department’s communications director, said earlier this week.