Marshall (7-13, 1-4 Conference USA) snapped a four-game losing streak and never trailed against Rice (5-13, 0-5).
The Thundering Herd scored the first eight points of the game en route to a 33-22 halftime lead. Rice opened the second half on a 13-5 run and came within 38-35 with 14:45 left.
Marshall took a 46-35 lead after an 8-0 run, highlighted by 3-pointers on consecutive possessions from Canty and then Taylor.
The lead was cut to 69-63 with 18 seconds left after a Marcus Jackson 3-pointer. But Canty and Taylor each hit a pair of free throws in the final 13 seconds to preserve the win.
Sean Obi led the Owls with 22 points and 17 rebounds. Max Guercy had 11 points and Jackson added 10.