The Tigers (17-7, 6-4 Atlantic Coast Conference) looked like they had this one wrapped up after Stitt's 3-pointer made it 59-42 with 8:22 to go.
But the experienced Eagles (15-9, 5-5) put together a 12-0 run to tighten things up. Reggie Jackson hit a 3-pointer, Joe Trapani added a three-point play and Biko Paris a 3 to cut Clemson's lead to 59-54 with less than 5 minutes left.
Jackson finished with 27 points and Trapani 22 for the Eagles, who fell to 0-5 all-time at Littlejohn Coliseum.
Young had 15 points and Smith 11 for Clemson, which moved to 13-1 overall and 5-0 in ACC play at home this season.
Stitt and Young got it going in the second half, combining for 26 of their 31 points after halftime.
Grant and the Tigers started the second period with a 20-5 run. Grant, the son of former Oklahoma standout Harvey, had a short jump shot, a jam and a three-point play during the surge.
Boston College could not recover.
Grant finished 7 of 9 from the field and 3 of 3 from the foul line. He also added three blocks and a steal.
The Tigers won despite getting outrebounded 36-29 by the taller, more experienced Eagles, who started three seniors and a junior. But Clemson committed just seven turnovers and hit 17 of 22 foul shots to hold onto their lead.
Neither team was expected to do much before the year began, both the Tigers in seventh and the Eagles in 10th landing in the lower half of the ACC's preseason poll. Instead, both have pulled off a few league surprises and have themselves in the mix for one of the four first-round byes with the conference tournament about a month away.
The first half played out like a pair of evenly matched teams who wouldn't let the other pull away.
The Tigers opened with a 10-3 run that pumped up the late-night Littlejohn crowd. But then Boston College answered with a 17-5 run highlighted by two Corey Raji 3-pointers that put the Eagles ahead 20-15.
Clemson recovered from a cold streak -- it went 9 minutes with just one field goal -- to take control at the end of the period.
Young, a regular starter who wasn't in the opening lineup because he was late to a shootaround this week, hit a 3-pointer at the 2:13 mark that regained the lead for Clemson, 26-24.
A minute later, Young got a steal and pushed the ball to Smith for a jam. Grant closed the half with another easy dunk and Clemson took a 30-27 lead into the break.