Atlantic-10 Conference player of the year Jordair Jett overcame a slow start to score 18, doing most of his damage while the fifth-seeded Billikens (27-6) were erasing a 59-45 deficit over the last five minutes of regulation.
With better free throw shooting, Saint Louis would have won without having to work overtime against a team playing its second game in three nights.
No. 12 seed NC State (22-14) beat Xavier in an opening round game in Dayton, Ohio, and fatigue very well could have been a factor in the Wolfpack missing 17 attempts from the foul line.
T.J. Warren scored 28 points for NC State, but the Atlantic Coast Conference player of the year made just six of his 14 free throw attempts. He missed one that could have tied it in the final minute of OT.
Warren fouled out with 27.9 seconds remaining, leaving Ralston Turner to try to rescue the Wolfpack with a desperation 3-point attempt that clanked off the rim as time expired.
Saint Louis advanced to the third round to play either fourth-seeded Louisville or No. 13 Manhattan in a Midwest Region matchup on Saturday.
The Billikens did it despite trailing by as many as 16 and going 12 of 26 from the foul line.
Warren scored at least 20 for the 19th consecutive game. Turner, who finished with 16, hit a jumper and three free throws during a 19-7 spurt that helped the Wolfpack open their biggest lead of the night.
Saint Louis whittled away at its deficit, eventually tying the game at 70 on Jett's driving layup with 18.8 seconds remaining. But the senior failed to complete a three-point play that would have given the Billikens their first lead since the opening minutes of the game.
It went to overtime when point guard Tyler Lewis was unable to get the ball to Warren and missed an open 17-foot jumper at the buzzer.
In winning, Saint Louis avoided a recent trend that seen No. 5 seeds struggle against No. 12 seeds.
Earlier Thursday, No. 12 seed Harvard took down fifth-seeded Cincinnati 61-57 in an East Region game played in Spokane, Wash. North Dakota State also knocked off No. 5 Oklahoma in OT as well.
There's been at least one No. 12 seed winning every season since 2008.
Saint Louis played its best basketball of the season during a 19-game winning streak, however the Billikens stumbled down the stretch and had lost four of five heading into Thursday night.
North Carolina State, by contrast, surged late to earn a spot in the NCAA field of 68.
With Warren leading the way, the Wolfpack beat Pittsburgh and Boston College to end the regular season, then knocked off Miami and Syracuse in the ACC tournament to give themselves a shot. A first-round victory over Xavier in Dayton, Ohio, sent them to Orlando to face Saint Louis on one day's rest.
The short turnaround wasn't a hindrance.
Warren sparked a 15-4 run that turned an early 9-4 deficit into a 19-13 lead that the Wolfpack gradually built to 11 points. St. Louis got a long 3-pointer from Jake Barnett and a layup from Dwayne Evans to end the opening half on a 7-0 spurt that enabled the Billikens to pull within 30-26 at halftime.
The Billikens won their opening game in the tournament each of the past two seasons before losing third-game matchups against Michigan State in 2012 and Oregon a year ago.
N.C. State was in the tournament for the third straight year under coach Mark Gottfried, one of seven active coaches who've led three different program to multiple appearances in the NCAAs.
After beating San Diego State and Georgetown to make a surprising run to the round of 16 in 2012, the Wolfpack lost its opening game to Temple a year ago.