The Tigers went into the decisive game five having surrendered a 2-0 series lead, but ace Verlander steadied the ship with a typically dominant performance, allowing four hits and one walk while striking out 11.
Austin Jackson led the Detroit offence with two runs and two RBI on a pair of hits.
They will next meet either the New York Yankees or the Baltimore Orioles, who sent their ALDS series to game five in dramatic fashion with a 13-inning 2-1 victory.
JJ Hardy's RBI double proved to be the decisive moment in a marathon clash in New York as the Orioles tied the series up at 2-2.
Buster Posey's grand slam put San Francisco into the National League Championship Series as the Giants beat the Cincinnati Reds 6-4 to complete a remarkable series turnaround.
All San Francisco's runs came in the fifth inning off Mat Latos, topped off by Posey's slam.
It made the Giants the first National League team to rally from two games down to win a Divisional Series 3-2.
They still do not know whether they will face Washington or St Louis after Jayson Werth hit a walk-off homer to give the Nationals a 2-1 win over the Cardinals and send their NLDS series to game five.
It means all four divisional series have needed a fifth game.
Adam LaRoche also went yard for the Nationals.