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.