"Early on, didn't have great control of any of my pitches," Verlander said. "But I was able to get myself out of jams that I created."
Joaquin Benoit pitched the eighth and Jose Valverde struck out two in a perfect ninth for the save.
Oakland's Jarrod Parker allowed two earned runs in 61/3 innings and took the loss.
Game 2 is Sunday, with Doug Fister taking the mound for Detroit and lefthander Tommy Milone for Oakland.
After winning their final six games to take the AL West in shocking fashion, the A's made their presence felt right away in Detroit. The home crowd greeted Verlander with a roar and a sea of twirling white towels when he popped out of the dugout and headed to the mound to start the game, but Crisp was unfazed. He pulled Verlander's two-strike fastball just inside the pole in right field to put Oakland on top.
The AL Central-champion Tigers tied it immediately. Austin Jackson's hard-hit ball deflected off diving shortstop Stephen Drew and into short left field. The Detroit leadoff man ended up with a double and went to third when Quintin Berry slapped a single to third off Josh Donaldson, who also could get only a piece of the ball while diving for it.
Triple Crown winner Miguel Cabrera, who went hitless, grounded into a double play, but Jackson came home to make it 1-all.
Drew finally made a diving play in the second, sprawling to his right on Delmon Young's grounder and then throwing to first for the out. At the plate, the A's made Verlander work, forcing him to throw 61 pitches in the first three innings. The Detroit ace struck out Brandon Moss to end the Oakland third with a 99 m.p.h. fastball - but Verlander was having to reach back for extra speed early.
"Most good starters, you try to get to them before they get into their rhythm," Oakland manager Bob Melvin said. "He got better as the game went along."
The A's tied a postseason record by starting four rookies - Parker, Donaldson, Yoenis Cespedes, and Derek Norris.
Parker looked sharp early but allowed another run in the third because of a fielding mishap. With two out and a man on second, Berry chopped a soft grounder to the right side. Parker came off the mound to field it, but with the speedy Berry hustling to first, Parker lost control of the ball with his glove hand for an error that allowed Omar Infante to score.
Avila's solo shot made it 3-1, and the A's couldn't break through against Verlander.