Avila's previous homer came on July 10, when Detroit beat Minnesota to take over first place in the AL Central. In between, the Tigers went 7-24 to fall 10 1/2 games behind the Twins.
Devastated by injuries to several top players and struggling in all phases, the Tigers now find themselves desperate for victories.
"I've been around this game long enough to know we've got to start to put something together," said Brandon Inge, who led off the ninth with a single. "This one was huge."
Unlikely, too. The White Sox had been on a roll -- winning 41 of their previous 59 games -- and Avila was batting .210 before connecting on the first pitch he saw from Putz.
"When Avila hit that ball, it was, like, 'Come on!' Then, 'Really?"' said Phil Coke (7-2), who pitched 1 1/3 scoreless innings to get the victory.
Avila, who platoons at catcher with Gerald Laird, added: "It's been a really tough year for me personally, so it felt good. I was so excited that I was clapping as I went around the bases."
Putz (5-4) was working the ninth inning because Bobby Jenks is out with back spasms. Putz also gave up a game-ending homer to Baltimore's Brian Roberts on Monday.
Putz had seasons of 36 and 40 saves with Seattle, so he knows what it's like to pitch with the game on the line.
"It was away and [Avila] put a good swing on it and hit it the opposite way," said Putz, who earlier this season set a franchise record with 27 straight scoreless appearances. "If you're going to get beat, that's the way you want to get beat. I didn't think it was a terrible pitch. But that's the way it goes sometimes."
The White Sox have lost six of nine games to go from leading the division by 1½ games to trailing Minnesota by two.
Chicago pitcher Edwin Jackson struck out 11 in seven innings in his home debut, allowing one run and five hits. He has given up only three earned runs in 20 innings since he was acquired from Arizona on July 30.
"Another good start," White Sox manager Ozzie Guillen said. "Three out of three. You can't be better than that."
Rick Porcello pitched seven innings for Detroit, yielding two runs and nine hits. A rookie standout last season, Porcello struggled so badly earlier this year he was sent to the minors for a spell.
Porcello escaped serious damage by inducing three double-play grounders. With two on in the third inning, he was saved by second baseman Carlos Guillen, who ranged well behind the bag to grab Alex Rios' grounder and made a behind-the-back flip to shortstop Ramon Santiago to start a sensational DP.
"Best I've seen," Porcello said. "I was jumping up and down."
Detroit tied it at 1 in the sixth on Guillen's 425-foot homer to right-center field. It was the first homer since June 20 for Guillen, who had been 1 for 16 since coming off the disabled list Monday.
Even with the victory, the Tigers are 18-36 on the road and 3-7 against the White Sox.
- Tigers RF Brennan Boesch struck out four times. After a hot start to his rookie season, he is batting .125 over his last 29 games.
- Jenks isn't expected to pitch in Sunday's series finale, either. After Chicago is off Monday, the team will decide if Jenks needs to go on the DL. "I don't think he'll be ready by Tuesday," Ozzie Guillen said. "I hope he is. But you can't even move today and all of a sudden Tuesday you're fine?"