Villanova (10-2) won its fifth straight game and picked up its first playoff victory since 2002, when it advanced to the semifinals before losing to McNeese State.
Jordan Scott scored three touchdowns for Patriot League champion Colgate (9-3), which had its eight-game winning streak snapped. It was Colgate's first loss since falling to Furman 42-21 on Sept. 13.
Szczur gave the Wildcats a quick 6-0 lead 13 seconds into the game.
Before Joe Marcoux's PAT attempt missed wide right, the game was delayed nearly 15 minutes as Villanova safety Darrel Young was carted off the field in an ambulance with a neck sprain. He was taken to University of Pennsylvania Hospital, and after the game Villanova coach Andy Talley said all tests came back negative.
Colgate eventually cut the lead to 13-7 when Scott scored on a 2-yard run with 1:22 left in the first quarter.
The Wildcats responded with 28 unanswered points for a commanding 41-7 halftime advantage. Four Villanova players scored in the second quarter.
Aaron Ball's 8-yard touchdown run for Villanova came with 1:41 remaining for the final score of the first half. Ball also scored from 6 yards in the first.
Szczur added a 4-yard run in the second and a 16-yard touchdown reception on a pass from Chris Whitney early in the third.
Scott, who rushed for 94 yards on 27 carries, set the FCS record for career carries with 1,240. He needed only three carries to break the previous all-time Division I record of 1,215 held by Steve Bartalo of Colorado State.