With 63 percent of the vote counted, Coleman led Mondale 51 percent to 46 percent.
Mondale entered the race last Thursday, an emergency replacement for Sen. Paul Wellstone who died in a plane crash just 11 days before Tuesday's election.
"The wave is moving from east to west and we're waiting for it to hit Minnesota," Coleman told cheering supporters. "When you see the energy in this room you know the future is in and with the Republican Party."
Mondale also urged his supporters to be patient.
"We've got a good chance," he said.
The whirlwind campaign was highlighted by Monday's debate between the two candidates. Mondale often sounded like he was lecturing the younger Coleman, referring to him as "Norman" while Coleman used a more deferential "vice president" in talking about his opponent.