Torii Hunter drove in three runs for Los Angeles, which reduced its magic number to clinch the AL West to eight.
Cabrera hit his third home run in three days and drove in three runs for Detroit, which has lost seven of nine.
With the score tied at 4, Fernando Rodney (0-4) walked Figgins to lead off the ninth.
"Walking Figgins was the kiss of death," Tigers manager Jim Leyland said.
Figgins then made a prophet out of Leyland by stealing his third base of the game. Figgins went to third on Garret Anderson's groundout, and Teixeira's fly ball to center made it 5-4.
"I just think its playing baseball," Figgins said. "My goal is to get on base, try and steal bases and score runs and let the big boys take care of business."
Cabrera hit his 31st homer with two outs in the eighth off Scot Shields (6-4) to tie the game.
Cabrera, who has nine RBI in his last three games, has driven in 109 runs. That ties him with Minnesota's Justin Morneau for second in the AL.
Cabrera's homer was the first run the Angels' bullpen allowed since Aug. 22, a span of 29 2/3 innings.
"They've been great," Los Angeles manager Mike Scioscia said. "We've obviously looked at a lot of guys and see an immediate impact every day, especially our late inning relief."
"The Angels always find a way to get him into the game. He wants the ball," Granderson said. "Even when he gets in trouble like tonight, he find a way to get out of it."
Hunter gave Los Angeles a 2-0 lead in the first with a two-run double with the bases loaded.
Cabrera's two-out RBI single in the bottom half scored Granderson, who led off with a single and stole second.
Vladimir Guerrero doubled with one out in the third and scored on Hunter's RBI single.
The Tigers tied it at 3 in the third. Angels starter Joe Saunders hit Placido Polanco with a pitch with one out and Magglio Ordonez followed with a double. Cabrera brought Ordonez home with a double off the right-field fence.
Saunders allowed three runs and eight hits in 6 1/3 innings, walking one and striking out two.
Tigers starter Chris Lambert gave up three runs and six hits in four innings.
Detroit LHP Dontrelle Willis and RHP Freddy Garcia pitched against each other in a simulated game before Tuesday night's contest. Leyland was pleased with both pitchers' progress. ... Willis spent most of the season at Class-A Lakeland working on his control and mechanics and was then sent to Triple-A Toledo. The Tigers recalled him on Monday. Garcia, who hasn't pitched in the majors since mid-2007 was signed to a minor league contract and is not on the active roster. ... Los Angeles recalled C Ryan Budde and RHPs Dustin Moseley and Darren O'Day from Triple-A Salt Lake City. ... Detroit recalled the contracts of RHP Chris Lambert and LHP Clay Rapada from Triple-A Toledo and purchased the contracts of C Dusty Ryan and INF Mike Hessman from the same team on Tuesday.