Mike Trombley (3-3), who has an 8.83 ERA in 34 2/3 career innings against Cleveland, walked David Bell on a 3-2 pitch leading off the ninth. Kenny Lofton, who had struck out three straight times, singled to right off Swindell to send pinch-runner Shawon Dunston to third with nobody out.
Swindell walked Omar Vizquel intentionally to load the bases for Justice, who gave Cleveland its seventh last at-bat victory at home with a ground single to right past a drawn-in infield.
Mike Jackson (1-1) pitched a perfect ninth for the victory after Alvin Morman, Eric Plunk and Paul Assenmacher combined to blow two leads. Bartolo Colon, the winning pitcher in the All-Star Game, got a no-decision and had his five-game wining streak snapped.
Minnesota's Ron Coomer was 4-for-5 for his fourth career four-hit game, including his 12th homer.
Cleveland's Manny Ramirez drove in the tiebreaking run with a bloop single in the seventh, only to watch Brent Gates' RBI single fall in front of him to tie it at 5 in the eighth.
Ramirez's bloop to left off Trombley scored Justice, who walked with two outs against Eddie Guardado. Jim Thome, who followed Justice with a walk, was thrown out at third to end the inning.
Plunk, who gave up Orlando Merced's RBI single that tied it at 4 in the seventh, walked Pat Meares leading off the eighth. With two outs, Gates singled to right off Assenmacher to tie it again.
Travis Fryman hit his 15th homer and drove in three runs, giving the Indians a 3-2 lead with a two-run shot off starter LaTroy Hawkins in the fourth. Hawkins allowed four runs and seven hits in five innings.
Colon allowed three runs and eight hits in 5 2/3 innings. He walked four and struck out three, the first time he's walked more than he's struck out.
The Twins took a 2-0 lead in the first on an RBI double by Matt Lawton and a run-scoring groundout by Merced.
Fryman cut it to 2-1 with a sacrifice fly in the second, then followed Sandy Alomar's bloop single with a 411-foot, two-run homer in the fourth to give Cleveland a 3-2 lead.
Ramirez made it 4-2 with a RBI single in the fifth, scoring Thome, who doubled. Coomer cut it to 4-3 with a leadoff homer in the sixth against Colon.
The Twins had three runners thrown out trying to advance on the bases, including Gates at the plate in the third. He tried to score from second on a failed 3-6-3 double-play attempt and was thrown out by Thome.
The Indians had Thome cut down at third in the seventh and Kenny Lofton picked off first in the first.