Snider came up with a loose ball rebound on the defensive end of the floor and was fouled before going to the line and sinking both free throws.
Terry Rozier then came up with a steal at midcourt and the Cardinals edged UC-Irvine 57-55 before 14,509 fans at KeyArena.
"He stepped to the line, as a freshman, and made two big free throws," U of L coach Rick Pitino said. "We're very excited to move on."
The Cardinals (25-8) will play Northern Iowa - a winner over Wyoming - on Sunday afternoon. The game times will be announced later tonight.
The fourth-seeded Cardinals could never shake the Anteaters. With the game tied, Snider snagged a rebound in the corner off a long miss by Luke Nelson and drew a foul.
He calmly walked to the line and sank both free throws. The Cardinals had two fouls to give and after a timeout by Pitino used one of the fouls near midcourt.
Rozier then went for the ball with about four seconds left, tipped it away and it rolled off a UC-Irvine players leg to Wayne Blackshear as time ran out.
"No foul," Rozier said. "I told (Montrezl Harrell) that I was going for the steal. Coach had said foul, but I thought I could get the steal."
Blackshear led the Cardinals with 19 points and now has 998 for his career. Snider had a career-high 16 points and five rebounds in 37 minutes. Rozier added 12, while Harrell had eight points and four rebounds.
Mamadou Ndiaye, the 7-foot-6 center and tallest player in college basketball, finished with 12 points and five rebounds for the Anteaters.
U of L hit 23 of 55 from the field and was 6 of 6 from the free-throw line.