Jalen Duren added 10 points and 11 rebounds as the Tigers (22-10) won for the 13th time in the last 15 games in their first Tournament appearance since 2014.
"I just felt like he (coach Penny Hardaway) did a great job of preparing us for that moment", he said. "Just throughout my past and my history, i've always been in games where I could be up 20 or down 20 and the game is never over.
"Coach Penny put us in the right spots that always helps get through every time."
Abu Kigab was the high man for Boise State (27-8) with 20 as the Broncos, Mountain West regular-season and tournament champions, fell to 0-7 all-time in the NCAA Tournament.
"That's a team that's putting it together", said Boise State head coach Leon Rice of Memphis. "I know they had a rough stretch and I know they started out top ten in the nation and you see why.
"But they worked through some stuff throughout the year and they've turned it on right now and they've got it going."
Story of the game
Memphis, who reached the final of the American Athletic Conference tournament last week, went on an 8-0 run early to take a 27-17 lead with 4:26 left in the first half after a three-pointer by Emoni Bates.
Alex Lomax then connected from long distance a few minutes later that gave Memphis a 38-19 lead at the break, but he came down hard and appeared to injure his left ankle.
Boise State came out strong to start the second half, using a 16-4 run to narrow the gap to 42-35 with 9:45 left in the game.
Williams' dunk seemed to spark the Tigers again to make it 49-40 with 9:18 left. Kigab hit a three and a short jumper to pull Boise State to 49-43 with 5:55 remaining, but Williams answered from long range.
The Broncos closed trimmed the deficit to 56-51 with 1:26 left on a layup by Naje Smith and the teams then traded foul shots. Duren's layup pushed the margin to 60-53 to give Memphis some breathing room.
Lomax came back to ice the game with some late free throws.