Cabrera went 2-for-4, with a 4th inning double that plated two runs and a 6th inning homer with Dirks aboard. The home run was Cabrera's 26th of the season. He had four RBI to up his total to 92.
Avila, who caught the whole game, was 4-for-6, with a pair of ribbies and two runs.
The Tigers swept the weekend series from the White Sox, winning the first two games, 8-1 and 9-8, sending Chicago back down to third place in the AL Central, 8 1/2 games back, while increasing their lead over second-place Cleveland to 6 1/2 games.
Milestones: Toronto's Jose Bautista hit his 40th home run of the season to lead the majors. Adrian Beltre collected his 2000th hit in the Rangers' win over Boston.
It was a little different story over in the NL, where Shaun Marcum walked one and allowed just one hit over seven innings as the Brewers continued to demolish the competition with a 4-0 win over Houston.
Ryan Braun provided the bulk of the offense with a solo homer in the 1st, a bases-loaded walk in the 4th and a run-scoring single in the 7th inning. He went 3-for-3 with three RBI. Braun is having an MVP-like year, batting .325 with 26 home runs and 94 RBI.
The win dropped Milwaukee's magic number for clinching the division to 13. They have a 9 1/2-game lead on St. Louis.