It was the 10th win in 11 games for Memphis, which became the sixth NBA team this season to reach 40 victories.
The Grizzlies overcame a 12-point halftime deficit and took the lead for good on a jumper by Tayshaun Prince with 4 minutes remaining.
Portland found itself just two points behind after a layup from Damian Lillard with 17 seconds left, but Conley and Allen came through at the foul line to clinch the victory.
Marc Gasol led Memphis with 23 points and 12 rebounds. Conley had 20 points and Jarryd Bayless added 13.
Lillard scored 20 to pace the Trail Blazers. Portland had won its previous two games, but shot just 31 percent from the field in this one.