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.
SMU has signed a second player from Australia, 6-foot-11 forward Harry Froling, to a national letter of intent. The Mustangs previously signed guard Tom Wilson from Australia.
There's nothing to suggest the Jayhawks won't end up atop the Big 12 for a 13th season in a row. But what about a return to the Final Four?
Colorado State graduate transfer John Gillon has chosen Syracuse, giving the Orange a potential starter at point guard entering the 2016-17 season.