Nate Wolters scored a team-high 14 points in the victory. Callahan finished with 10 points and six rebounds and also made a pair of free throws with 15 seconds remaining to force overtime.
The Jackrabbits (27-7), who advanced to the title game with a 63-47 win over Southern Utah on Monday, will make their 25th trip to the NCAA Tournament, but their first at the Division 1 level.
Tommie Tyler scored a game-high 19 points in defeat. Jack Houpt had nine points and seven rebounds and Terell Parks chipped in eight and 11 for the Fighting Leathernecks (18-14), who knocked off top-seeded Oral Roberts on Monday.
Western Illinois was in control early.
It scored five straight to close the first half, turning a 24-23 deficit into a 28-24 advantage in the process.
Carrying the momentum out of the locker room, the Leathernecks opened the second half with a 9-1 push.
Tyler scored four points during the run, capping it with a jumper to make it 37-25, marking the largest lead of the game for either team.
The Jackrabbits did not pull even until Callahan's free throws with 15 seconds to go. In fact, they were still down nine, 42-33, with 7:35 to play before closing regulation with an 11-2 surge to send the game into overtime.
A layup by Obi Emegano had Western Illinois up, 50-49, with two minutes to play in the extra session. But Callahan grabbed an offensive rebound off a Wolters miss and knocked down the go-ahead three on South Dakota State's next possession.
Tyler missed a short shot just before the final buzzer sounded.
South Dakota State has won seven straight over Western Illinois and leads the all-time series, 9-3...The Jackrabbits will be making their first trip to the NCAA Tournament at any level since the 2003-04 season.