Redford, previously scoreless for the game, was fouled by Lucky Jones after a ball fake in the corner and made both free throws. The Colonials (1-3) called two timeouts after regaining possession, but Karvel Anderson missed a 3-pointer from the left wing with 12 seconds left.
Robert Morris maintained possession after a held ball, but Velton Jones couldn't connect on a leaner and Coron Williams missed a desperation 3-pointer as he was falling out of bounds.
Justin Martin scored a career-high 17 points and Dee Davis added 15 for the Musketeers (3-0).
Freshman Semaj Christon, who set Xavier's single-game freshman assist record with eight in the Musketeers' 62-47 win over Butler on Tuesday, finished with 12 and a team-high six assists in just his second career game and first start. Christon missed Xavier's opener while recovering from an infection.
Martin, who scored a combined 21 points in Xavier's first two games, was 6 of 9 from the field, including a career-high four 3-pointers in six tries.
The Musketeers overcame a season-high 15 turnovers to go 3-0 for the fifth straight season and improved to 4-0 in their series against Robert Morris (1-3).
Anderson led the upset-minded Colonials with a career-high 17 points. Velton Jones added 15 for Robert Morris.
Xavier led by as many as eight points, 43-35, on a Jeff Robinson dunk with 15:48 left in the game, but the Colonials came back to tie the score twice in the final 3:49, the second at 59-59 on a jumper by Velton Jones with 1:51 left in the game.
The Colonials, who made just 16 of 70 3-pointers over their first three games, sank their first five and 6 of 9 in the first half, and led 19-17 with 12:28 left before halftime. Xavier countered by shooting a sizzling 65 percent (13 of 20) from the field to lead by as many as seven, 26-19, with 8:31 left before settling for a 36-31 halftime advantage on Davis's 3-pointer with 1 second left.
The Musketeers never trailed after Christon's driving layup gave them a 21-19 lead with 11:01 left in the first half.
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