Marquette had too many weapons against the outmanned Wildcats (18-13), who played without leading rebounder and No. 2 scorer Patrick Patterson. Kentucky went 2-3 after losing the star freshman to a season-ending ankle injury.
As it turned out, Crawford was almost enough for the Wildcats.
Kentucky cut it to 66-63 on Crawford's 3-pointer from the top of the key with 38 seconds left.
After Matthews hit two free throws, Bradley hit a 3-pointer from the corner to get the Wildcats within two with 22 seconds remaining.
Matthews made two more free throws, then stole the ball and hit two more from the line to ice the game.
Crawford hit 13 of 22 shots from the floor while his teammates went 10-for-26. He hit 5-of-8 from beyond the arc and 4-of-8 at the line.
Kentucky trailed 33-29 at halftime, but Crawford opened the second half by scoring the Wildcats' first seven points, on a 3-pointer, a putback and a layup to tie it at 36-36.
That's when the Golden Eagles awoke.
They went on a 9-2 run, sparked by Lazar Hayward's thunderous dunk off a feed from McNeal.
Six minutes into the half, Marquette took its biggest lead - 45-38 - when McNeal scooped up a blocked shot and raced the length of the court for a layup.
Kentucky trailed 54-43 midway through the second half when Crawford went on a personal 6-0 run, scoring on a three-point play and then hitting from beyond the arc to pull the Wildcats within five points.
The Wildcats and Golden Eagles met for the 10th time in the NCAA tourney's most-played matchup. Marquette is 6-4 in the series.
Both tradition-laden teams were hitting their stride as the postseason began.
Marquette opened 6-5 in the Big East but went 8-3 overall down the stretch, losing to eventual champion Pittsburgh in the Big East semifinals. Kentucky overcame a 7-9 start, including a loss to Gardner-Webb, and won 11 of its last 14 to slip into the bracket.
Bradley and Crawford combined to score 21 of Kentucky's 29 first-half points.
Patterson's loss was most evident under the boards. He led Kentucky in rebounding, and Marquette exploited his absence, outrebounding Kentucky 21-15 in the first half.
Marquette attacked the basket from the start, scoring 14 of its first 17 points in the paint.
Marquette went on a 12-0 run midway through the first half to take a 16-10 lead. But the Wildcats rallied to tie it at 23 on a fastbreak dunk by Crawford off an alley-oop pass from Bradley.
KENTUCKY (18-13): Stevenson 4-6 2-2 10, Crawford 13-22 4-8 35, Harris 0-0 0-0 0, Bradley 6-15 4-4 19, Jasper 0-3 0-0 0, Porter 0-1 2-2 2, Stewart 0-1 0-0 0, Perry 0-0 0-0 0, Coury 0-0 0-1 0. Totals 23-48 12-17 66.
MARQUETTE (25-9): Hayward 7-11 0-0 16, Barro 1-3 2-2 4, James 3-9 8-10 15, McNeal 8-15 2-2 20, Matthews 2-10 9-10 13, Acker 1-3 0-0 2, Fitzgerald 1-1 0-0 2, Cubillan 0-0 1-2 1, Burke 0-0 1-3 1. Totals 23-52 23-29 74.
Halftime-Marquette 33-29. 3-Point Goals-Kentucky 8-20 (Crawford 5-8, Bradley 3-9, Porter 0-1, Jasper 0-2), Marquette 5-13 (Hayward 2-4, McNeal 2-5, James 1-3, Matthews 0-1). Fouled Out-Crawford, Harris. Rebounds-Kentucky 26 (Jasper 7), Marquette 34 (Hayward 7). Assists-Kentucky 13 (Bradley, Jasper 4), Marquette 10 (James 3). Total Fouls-Kentucky 22, Marquette 18. A-NA. . A-NA.