The Grizzlies extended New Jersey's losing streak to seven games.
New Jersey took a 93-92 lead on two free throws from Kris Humphries with 3:18 to play but Allen put Memphis in front for good 19 seconds later by hitting a 3-pointer.
An Allen layup with 1:32 to go extended the lead to 99-93 and the Nets came no closer than four the rest of the way.
Rudy Gay scored 25 points to lead Memphis, Allen had 21 and Marreese Speights had a career-best 18 rebounds to go with his 20 points.
Deron Williams led New Jersey with 26 points and 11 assists. The Nets' losing streak is their longest of the season. New Jersey has won only three home games all season, second-fewest in the league.