The Rams won their third game in five days, victories that have come after they were widely denounced as being unworthy of appearing in the tournament.
Virginia Commonwealth finished fourth in the Colonial Athletic Association regular-season race and then lost in the title game of the league's tournament to Old Dominion.
Since then, however, the Rams have defeated USC by 13 points, Georgetown by 18 and Purdue by 18. As one of the last four at-large teams to receive a tournament bid, the Rams had to play an extra game just to get into the main bracket of 64.
Virginia Commonwealth, seeded 11th in the Southwest Region, will next take on No. 12 seed Florida State in San Antonio. It will be the Rams' first trip to the regional semifinals and they will meet a team that is seeded 10th.
Burgess was one of six Rams in double figures, helping fuel a run during the final 6:30 of the first half that turned a one-point deficit into a 10-point lead at the break. That advantage grew to 19 points during the first 8:30 of the second half.
Joey Rodriguez added 12 points with 11 assists and no turnovers for the Rams (26-11). Jamie Skeen had 13 points and Juvonte Reddick 12.
Purdue (26-8), which finished second to Ohio State in the Big Ten Conference, was led by 25 points from JaJuan Johnson. The Boilermakers had been trying for their third straight Sweet 16 appearance.