Mauer lined a one-out double off the wall in left before Willingham connected for his 15th homer, a 438-foot drive into the second level in left.
"It was fun to watch Joe's at-bat. It set the tone for the inning," Willingham said. "I was looking for something over the middle hard and the pitch was over the middle and it was hard. I tried to stay short with my swing. You don't swing early but just try to make it shorter."
Chapman (4-4) yielded a game-ending, two-run homer to Asdrubal Cabrera in his previous outing, a 3-2 loss at Cleveland on Tuesday. The lanky left-hander is 0-4 with three blown saves and an unseemly 11.37 ERA in his last seven games.
Willingham's homer came after Joey Votto hit a two-run shot to put the Reds in front in the eighth.
"If you need a fastball hitter in those situations, Josh is the guy," Twins manager Ron Gardenhire said. "Mauer's a great hitter. He can do things that others can't against a pitcher who throws above the speed limit."
Chapman struck out Darin Mastroianni to start the inning, then threw two quick strikes to Mauer. But Mauer worked the count full before coming up with the big hit.
"I tried to stay short. It's tough in interleague because you haven't seen the pitchers before and you don't know how the ball moves," Mauer said. "The guy's got a heck of an arm and has good movement on his pitches. I was trying to put the barrel on the ball. A guy that throws that hard supplies most of the power."
Scott Diamond (6-3) allowed three runs and eight hits in eight innings for Minnesota. The left-hander struck out seven and also hit two batters.
The home run by fellow Canadian Votto was disheartening to the rookie.
"I felt I made a good pitch," Diamond said. "From that pitch to Willingham going off it was pretty suspenseful but we remained positive."
Burton spent the last five seasons in Cincinnati. He opted for free agency when the Reds wanted to send him to the minor leagues last September; then signed with Minnesota two months later.
"The last out was pretty intense," he said. "I had a little extra adrenaline going but I tried to tell myself the pressure was on the hitter because they were the ones losing."
Burton walked Todd Frazier to start the ninth. Frazier moved up on Willie Harris' sacrifice and Burton walked pinch hitter Scott Rolen before Chris Heisey flied out on the first pitch and Wilson Valdez bounced into a fielder's choice.
"I had to be careful with Rolen," Burton said. "He's a professional hitter. I felt like I made good pitches the rest of the inning."
Mike Leake pitched eight solid innings for Cincinnati, surrendering just five hits. He struck out five, walked none and had just 86 pitches when manager Dusty Baker went to Chapman for the ninth.
"That's Chapman's job," Baker said. "Leake just came off throwing 112 pitches his last time out. Everybody's got a job. What if I send Leake out and he gives it up? The hitters coming up would have seen him for the fourth time."
The Reds dropped to 7-8 in interleague play this season and 104-131 all-time. The Twins finished 9-9 this year and are 157-125 all-time.
The NL Central leaders grabbed the lead in the fourth. Devin Mesoraco hit a leadoff double, moved up on Leake's sacrifice and scored on Wilson Valdez's two-out single.
Trevor Plouffe responded with his 15th homer, a 385-foot drive to right-center that tied it with one out in the fifth.
There has been at least one homer at Great American Ball Park for 62 consecutive games, the longest current streak in the majors.
Justin Morneau's run-scoring groundout put the Twins in front in the seventh, but Votto's 14th homer made it 3-2 in the eighth.
Capps will be examined on Monday after the Twins return from their road trip. The team was discussing possible roster moves, Gardenhire said. ... Minnesota definitely will add a pitcher as a 26th player for a scheduled doubleheader Saturday against Kansas City, Gardenhire said. ... Minnesota CF Denard Span, hitless in his last 10 at-bats, was out of the starting lineup. Ben Revere moved from right field to center. ... Reds SS Zack Cozart, who had one hit in 14 at-bats over his last three games, rested. Valdez started in his place.