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.
Colorado State graduate transfer John Gillon has chosen Syracuse, giving the Orange a potential starter at point guard entering the 2016-17 season.
DeSean Murray, who averaged 20.2 points as a sophomore at Presbyterian, will transfer to Auburn to play for coach Bruce Pearl.
Former Kentucky guard Ed Davender, the only Wildcat with at least 1,500 points and 400 assists, has died. He was 49.