The Panthers (14-1, 2-0) shot 53 percent from the floor and pulled away from the Terrapins in the second half to win their first ACC home game.
Allen led Maryland (10-6, 2-1) but the Terrapins couldn't keep up with the Panthers.
Durand Johnson, limited by the flu in a victory at N.C. State on Saturday, keyed a late first-half surge that put Pitt in front and kept it going in the second half. The Panthers used a 15-7 burst to build a 13-point lead, then pulled away.
The Terrapins were looking for their first 3-0 start in ACC play since 2002, when they went on to win the national title behind Juan Dixon and Steve Blake.
Pitt used a lineup that spread the floor, giving Lamar Patterson and Johnson room to work. Patterson, who is developing into one of the ACC's best all-around players, kept the Panthers in it after the Terrapins got off to a hot start.
Maryland led 26-23 with 6 minutes left in the half when Johnson got rolling. The Baltimore, Md., native did his best to take it out on the Terrapins. Chastised at times by Pitt coach Jamie Dixon for his penchant for just chucking 3-pointers, Johnson hit a step-back jumper to get Pitt going then added a 3-pointer and finished the half with a smooth layup as Pitt took a 36-30 lead into the break.
Johnson, typically the first player off the bench, did enough to earn a rare second-half start from Jamie Dixon. It immediately paid dividends. He started the half with a layup in traffic then knocked down another wide-open 3.
Maryland turned it over 13 times to just seven for Pittsburgh and shot just 35 percent (20 of 56) from the field.
A pair of free throws by Allen pulled the Terrapins within 40-37 early in the second half but Maryland would get no closer. Pitt moved the lead to 55-44 on a 3-pointer by Josh Newkirk and the Terrapins never got within single digits the rest of the way.
Follow Will Graves on Twitter @WillGravesAP