Dabney made 6 of his 12 field goal attempts and finished a perfect 9 of 9 at the free throw line.
Malik Morgan had nine points, six rebounds and four steals for the Green Wave (6-4), who have won four of five games.
Tulane held Prairie View A&M to just 28.6 percent shooting and forced 20 Panthers turnovers.
With the game tied at 30 with 13:52 remaining, the Green Wave went on an 11-0 run to take their first double digit lead of the night.
Admassu Williams scored 13 for the Panthers.