The Bulldogs seemed in control owning a 62-55 lead with 3 minutes 15 seconds left following two free throws by junior guard Craig Stanley (Plano, Texas).
But Drake missed its last four shots. Two free throws by Evansville reserve James Haarsma with 17 seconds remaining tied the score at 62-62.
Then Stanley committed a turnover from Drake's right wing with 3.6 seconds remaining.
Following two time outs, Williams received the in-bound pass from Kavon Lacey and dribbled past mid-court before launching his game winning shot that hit the glass and went in.
The game was originally scheduled to be played Wednesday night but travel problems by Evansville forced officials to move the game to Thursday.
Evansville improved to 13-7, including a 5-5 league mark. Drake, dropping its second straight home game, fell to 14-8 overall with a 5-5 conference mark.
The Bulldogs opened the game on fire by making five of their first six shots to enjoy a 13-6 lead.
Junior guard Josh Young (Lawton, Okla.) scored Drake's initial eight points of the game.
Young led Drake with 16 points along with five rebounds and two steals.
A three-point basket by Adam Templeton (Rapid City, S.D.) gave Drake a 20-11 advantage which was the first of four nine-point leads in the first half.
Two free throws by Stanley with 4:20 remaining in the first half gave Drake its biggest lead of the first half at 33-23.
Drake expanded the lead to 11 points in the second half on two occasions with the latter at 47-36 following a layup by Jonathan Cox.
Cox finished with 14 points, nine rebounds two blocked shots and two steals.
Stanley added 12 points, while dishing out five assists.
Drake shot 50 percent in the first half en route to erecting a 38-30 halftime lead.
But the Bulldogs cooled off in the second half, hitting just 37.5 percent making nine of 24 shots.
Evansville scored 15 points off Drake turnovers, compared to the Bulldogs scoring just eight points off Evansville miscues.
Nate Garner led Evansville with 20 points, while Shy Ely chipped in with 18.