Democrats had argued earlier in the day that election officials' decision not to count ballots cast for Wellstone but to count those cast for his Republican opponent, Norm Coleman, put Democratic voters at a disadvantage.
The court issued its order without explanation. It also directed election officials to count only the most recent ballot received.
Wellstone and seven others died in a plane crash Friday, sending the election process into chaos. Democrats Wednesday chose former Vice President Walter Mondale to replace Wellstone on Tuesday's ballot.
Attorney Alan Weinblatt asked the court to take reasonable action.
"We have been told we ran out of luck," Weinblatt told the court during 75 minutes of arguments earlier in the day. "Your honors, we did run out of luck. It's true. We ran out of luck last Friday when an airplane went down. But we haven't run out of the law. We haven't run out of justice. We haven't run out of fairness."
Weinblatt asked the court to use "21st century tools" to give voters the opportunity to participate in Tuesday's election, including the Internet, faxes and e-mail.
"What I'm asking is different than what we've done before," Weinblatt admitted, but added it's necessary in the wake of the death of Sen. Paul Wellstone.
Attorney General Mike Hatch did not oppose a suggestion that substitute ballots be mailed to voters who requested them by phone. Earlier, he had said replacement ballots could be cast only in person.
The state printed 4 million ballots this year, all bearing Wellstone's name. It began printing supplemental ballots Thursday morning after Mondale's name was certified. Some 104,000 absentee ballots were sent out. Absentee voters who have not yet turned in their ballots and want to vote for Mondale have been advised to write in Mondale's name on the line provided.
While the ballot issue percolated, the campaign got into full swing.
Mondale agreed to debate his opponent once before the election and hit the campaign trail.
"I want to have a chance to listen to Minnesotans now about how they feel on these issues. And then I will debate," Mondale said. "I have authorized my assistants to pursue that. But, by all means, let's have a debate."
Republicans had been pressing for a series of debates but Mondale told Minnesota Public Radio one would have to suffice. He planned instead to hold a series of town hall meetings around the state.
Mondale's first campaign appearance was Macalester College in St. Paul, where he talked about Wellstone's legacy and his abbreviated Senate campaign.
"My opponent has been out here for six years. I've been in the race for 12 hours," he told a gaggle of reporters at Democratic-Farm-Labor Party offices in Minneapolis. That is the formal name of the Democratic Party in Minnesota.
Coleman resumed campaigning Wednesday, ending a four-day hiatus imposed by Wellstone's death and logging more than 800 miles on a fly-around. President Bush will join Coleman in Minneapolis Sunday on his fifth visit to Minnesota.