DeAndre Kane and Dago Pena added 14 apiece for the Thundering Herd (17-10, 8-5 Conference USA) who shot 44.6 percent (29 of 65) from the field.
Houston led throughout the first half and took a 31-23 lead into intermission. However, Marshall opened the second half on a 16-1 run to take a 39-32 lead with 13:26 left to play.
Marshall pushed that lead to 59-40 behind a jumper from Pitts.
Nigel Spikes came off the bench to grab 10 boards for Marshall in just 17 minutes as the Thundering Herd won the rebounding battle 43-40.
Marshall shot 57.6 percent (19 of 33) in the second half, after shooting 31.3 percent (10 of 32) in the first half, and forced 15 Houston turnovers.
Leon Gibson led Houston (12-14, 4-9) with 17 points.