Kansas (31-3) advanced to the regional semifinals, where it will face Illinois or Creighton on Friday in Madison, Wis.
The Jayhawks struggled with Holy Cross in the first round and were thought to be vulnerable with star guard Kirk Hinrich hampered by a sprained ankle.
But Collison scored for a 6-0 lead just 97 seconds into the contest. Freshman Aaron Miles added a pair of free throws and Collison made consecutive baskets to extend the margin to 12.
Boschee made a three-pointer for a 15-0 advantage with 15:46 left. Stanford (20-10) missed its first five shots and the Jayhawks blocked four in the opening stretch.
"I really hope that Kansas wins it all," Stanford coach Mike Montgomery said. "People are saying they're struggling at 31-3. Nobody does a better job than (coach) Roy Williams."
Hinrich entered the game with 13:05 remaining in the first half and had a basket and a pair of three-pointers to extend the margin to 31-15. He finished with 15 points and eight assists in 21 minutes.
"I don't know how to explain Kirk Hinrich to start off with," Williams said. "He felt 80 percent. I decided to give him a chance, but I didn't know how he was moving. He's just about as tough of a youngster as I've ever been around."
Stanford was still within 38-26 before Drew Gooden had a basket for Kansas. Freshman Wayne Simien scored six straight points and Hinrich beat the buzzer with a jumper for a 48-26 advantage.
"I didn't know even this morning if I could make it or not," Hinrich said. "Once I got in the game, however, it seemed to loosen up and really didn't hurt that much."
Gooden finished with 15 points and six rebounds for Kansas, which held a 46-36 advantage in points in the paint. The Jayhawks also had a 43-36 edge on the glass.
The Cardinal shot just 29 percent in the first half. After a physical game against Western Kentucky on Thursday, Stanford never had its legs and was just three of 22 from beyond the arc.
"After they lost to Oklahoma and almost got beat by Holy Cross, I thought, `Hey, we can beat those guys,'" Stanford forward Casey Jacobsen said. "They have a great team. I wish them all the best."
Williams had indicated he would not play Hinrich. He told the team to win without him to get Hinrich another game in the tournament.
Kansas just had too many weapons for the Cardinal, shooting 52 percent. Boschee drilled five three-pointers and Hinrich three.
"I was just so hyped for the game," Gooden said. "I was mentally focused. It was contagious and everybody caught on and went all-out."
Jacobsen scored 24 points and Curtis Borchardt added 13 and 11 rebounds for Stanford, which received minimal contributions from the rest of the team. The starting backcourt of Julius Barnes and Tony Giovacchini combined to make just two of 12 shots.