The Indians have now dropped a season-high eight consecutive games.
"We weren't able to open it up with a big inning," Grady Sizemore said. "To keep a team like that around long enough ... they are hot right now, they had the big inning, and that's all it was."
The Indians took an early lead with a second-inning RBI double by Kelly Shoppach off Twins starter Glen Perkins. The Tribe added to it in the sixth when Jhonny Peralta hit a two-run homer 391 feet over the left-field wall.
"We had them right where we wanted them," Lee said.
But the lefty could not hold the lead. He gave up one run in the sixth, via a Denard Span leadoff triple and subsequent Alexi Casilla RBI groundout, before sustaining substantial damage in the seventh.
Lee induced a Justin Morneau groundout to begin the inning, but then walked Craig Monroe and allowed consecutive singles to Delmon Young and Brendan Harris. With the bases loaded, Nick Punto hit an RBI single to center, scoring Monroe. Lee next walked Span, forcing in Young. Carlos Gomez grounded to Andy Marte in the next at-bat, and with the runners moving, the third baseman elected to go to first. Harris scored and the Twins led, 4-3, before Lee got out of the inning.
The Twins had only one extra-base hit in the contest, Span's triple, while three of Minnesota's seven hits didn't make it out of the infield. Minnesota entered play Sunday leading the American League in infield hits with 103.
"It's my job to throw strikes and keep the team in the game," Lee said. "Today, I kind of let it get away from us. We were in a good position to win the game right there and I gave them a couple runs, especially walking [Span with the bases loaded] -- that's unacceptable."
Perkins (5-2, 4.27) exited with the lead, and the Twins' bullpen did the rest. Matt Guerrier pitched a perfect eighth, while Joe Nathan secured his 25th save with a scoreless ninth.
"They have a good bullpen," Sizemore said. "Nathan is a tough closer; he's one of the best. You obviously don't want to see him in the game."
Lee (11-2, 2.43) went seven innings and gave up four earned runs on six hits and two walks while losing his first decision since May 18 at Cincinnati. He entered the game with a career 6-2 record and 3.77 ERA in 13 starts against Minnesota.
Minnesota has now won 18 of 21 while Cleveland has lost 14 of its last 18.
"When things are going good, the ball is bouncing the right way and you catch the breaks," Tribe manager Eric Wedge said. "When you scuffle as a ballclub, or even individually, then it goes the other way."
The Indians have been swept in consecutive road series against division rivals Chicago and Minnesota. They now head to Detroit for a two-game set to conclude the road swing.
"This road trip, we didn't win a ballgame, but we were in a position to win four of them," Wedge said. "We are just one play, one hit, away from being a much better ballclub."
The Indians have an off-day to regroup before taking on the Tigers on Tuesday. C.C. Sabathia is scheduled to make that start, but trade rumors continue to swirl throughout baseball and speculation is rampant that Sabathia might have already made his last start with Cleveland.
"I think mentally, they need to get away from it. ... We have a lot going on right now," Wedge said. "So hopefully, with a good off-day, we can put a lot of this behind us and take a positive step forward."