"We'll take any win, the way we are playing," Chicago manager Ozzie Guillen said.
The White Sox had lost six of nine and haven't won consecutive games since their season-high three-game streak almost a month ago.
Garcia (3-2) gave up two runs over six innings, going five starts without a loss for the first time since the end of the 2006 season.
"If my shoulder is OK, and it has been all season, I can still pitch," he said. "When I'm healthy, I can still put the ball wherever I want it."
After Garcia closed 2006 strong in his previous stint in Chicago, injuries led to him going a combined 5-10 with Philadelphia, Detroit and the White Sox the next three seasons.
The crafty right-hander looks as if he could be back, winning three straight games after giving up two runs in each start.
"He has surprised a lot of people by how good he has been," Guillen said.
Rick Porcello (3-4) allowed four runs on eight hits in seven innings.
"He couldn't get the ball down and they whacked it," manager Jim Leyland said.
Pierre led off the game with a walk and scored, hit a two-out double to give Chicago a two-run lead in its three-run fourth and dove to rob Gerald Laird of an RBI single that would have made it 4-3 in the inning.
"When you've got a runner on third, you go for the catch," Pierre said. "If you don't get it, so what. The guy is going to score if you play it on the bounce, and he'll score if you miss it."
Brandon Inge got to third with one out after putting the Tigers ahead in the second inning, but they continued to struggle scoring runners from third with one or fewer outs.
Garcia stranded Inge by getting Laird to ground out and Adam Everett to pop out.
Boesch led off the fourth with a single, stole second and advanced on a fly. He didn't get home because Ramon Santiago grounded out, and the rookie made a mistake by not running. Laird later lined out to Pierre.
Austin Jackson doubled in the fifth, advanced on a wild pitch and was stranded because Damon struck out and Miguel Cabrera lined out. Damon's bat slipped out of his hands when he swung and missed, getting fooled by a slider with knuckleball action.
"It looked like the ball disappeared," Damon said.
Leyland lamented that the Tigers didn't force Garcia to throw more strikes.
"We swung at too many balls on Garcia," Leyland said. "If you do that, he'll pick you apart."
Detroit entered the game with a .475 success rate of scoring runners from third with less than two outs, leading just three teams in the category, according to STATS LLC.
Damon said he and his teammates have to improve in those situations.
"That's our top priority," Damon said. "We've been really bad at it so far this year."
Inge had a homer in the sixth inning overturned by video review, which gave the umpiring crew a chance to see that his blast to left was about a foot outside the pole.
"Maybe now we'll get some bounces going our way for a change," Guillen said.
The Tigers, who won their previous two series at home against the Red Sox and Yankees, open a road trip Wednesday night at Oakland with an 8-11 record on the road after winning 14 of 20 at home.
"If we can play better on the road, the sky is the limit for this team," Damon said.
Chicago's Sergio Santos, J.J. Putz and Bobby Jenks each pitched an inning of scoreless relief. ... Boesch, who is hitting .387, extended his hitting streak to eight games with his 10th multihit game in his 20-game career. ... Chicago opens a homestand Wednesday against the Angels. ... Detroit's Carlos Guillen, who has been on the DL for almost a month with strained left hamstring, expects to start a rehab assignment this weekend for Triple-A Toledo. ... Ozzie Guillen didn't answer questions from the media before the game and joked about it with a clubhouse attendant. "I'm not talking, tell them, Curtis Granderson," Guillen said to a young man, who looked a little like the former Tigers outfielder.