The Grizzlies ousted the defending Western Conference champion Thunder in five games, winning four in a row against a team that wilted away after the loss of star Russell Westbrook to a knee injury.
Memphis will now take on either San Antonio or Golden State in the conference finals. The Spurs own a 3-2 lead over the Warriors and will try to conclude that series in California on Thursday.
The Grizzlies outscored Oklahoma City by 17 points in the second period to take a 12-point advantage at intermission and they still had a 12-point cushion with 4 minutes to play.
Oklahoma City then made a final charge that brought the Thunder within two when Reggie Jackson hit a 3-pointer with 14.3 seconds remaining.
Zach Randolph was quickly fouled and he gave the Thunder a chance when he missed both his free throw attempts.
Kevin Durant, however, missed a jumper for the Thunder with 5 seconds to go and Allen converted two foul shots with 3.3 seconds remaining to clinch the victory.
Randolph led the way with 28 points and 14 rebounds. Mike Conley added 13 points and Allen had 11.
Memphis shot just 37 percent from the floor, but held Oklahoma City to 36 percent and forced 14 Thunder turnovers.
Durant finished his season with a 21-point effort while Serge Ibaka had 17.