Jesse Morgan had 12 and Williams finished with 10 for UMass (9-3), which has won six straight games. The Minutemen also secured their seventh straight road victory, the program's longest road streak since the 1997-98 season.
UMass led 37-29 at intermission but watched Miami (5-7) open the second half on a 9-2 run, cutting the deficit to 39-38. From there, the teams battled through eight lead changes and five ties until Williams' jumper gave UMass the lead for good, 70-69, with 1:12 left.
Allen Roberts led Miami with 21 points.
Mike Brey is looking at former Notre Dame players to fill two assistant jobs that have opened up with the departures of Martin Ingelsby and Anthony Solomon.
Purdue's 2016-17 prospects got a jolt Wednesday when forward Caleb Swanigan announced he will return for his sophomore season.
Clemson's Jaron Blossomgame says he is withdrawing from the NBA draft and returning to school for a final season.