Price (12-10) was tagged for nine consecutive hits during a third inning in which New York scored all eight of its runs to avenge a 5-2 defeat in Tuesday's opener of this three-game set.
The 2012 AL Cy Young recipient gave up 12 hits in all before being removed with still none out in the third. Price, coming off a complete-game one-hit loss to his former Tampa Bay squad last Thursday, had worked at least six innings in 19 consecutive starts coming in.
Derek Jeter knocked in two runs during the frame and Jacoby Ellsbury and Brian McCann each finished with a pair of hits and an RBI to help the Yankees close within 2 1/2 games of Seattle for the AL's second Wild Card.
Greene (4-1) did his part as well, notching eight strikeouts and holding Detroit to two runs on five hits.
Victor Martinez had a solo homer for the Tigers, winners of three straight coming in, with Torii Hunter ending 2-for-5 with an RBI in the loss.
Price was able to get out of a 28-pitch opening inning unscathed and induced a double-play ball to work out of a jam in the second, but fell completely apart as the Yankee bats erupted in the third.
Ellsbury began the barrage with a leadoff single and Jeter sent him home with a double to right for the game's first run. Martin Prado continued the hit parade with a single before Mark Teixeira socked an opposite-field double to give New York a 2-0 edge.
Carlos Beltran and McCann followed with singles to bring in two more runs, with a base hit from Chase Headley preceding back-to-back bases-loaded RBI singles from Brett Gardner and Francisco Cervelli that extended the margin to 6-0.
Price was then mercifully relieved by Blaine Hardy, who surrendered consecutive sacrifice flies to Ellsbury and Jeter that capped the eight-run outburst.
Greene, meanwhile, didn't permit a hit until Hunter singled to begin the bottom of the fourth in front of Miguel Cabrera's double into the left-field corner that got Detroit on the board.
Martinez smacked his 26th homer of the season in the sixth and scored the Tigers' third run two innings later, drawing a walk and coming around on Nick Castellanos' triple.
Hunter plated Ian Kinsler with a two-out single in the ninth to close out the scoring.
Notable deaths of 2014 [PHOTOS]