Gonzaga (32-2) led by 11 in the final 10 minutes but Southern stubbornly fought back to 59-58 before Pangos hit a 3-point shot from the right corner to give the Bulldogs breathing room at 62-58.
Southern missed two chances to get closer and Pangos was fouled. His two free throws proved to be the game's final points.
Gonzaga will go against ninth-seeded Wichita State in Saturday's third-round. Southern ends its season 23-10.
Gonzaga entered the tournament as the USA Today Coaches' poll No. 1 but Southern, behind Derick Beltran, gave the Bulldogs more of a game than expected. Beltran had seven straight points for Southern in one stretch and it was his jumper with 3:47 to play that tied that score at 56-56.
Gary Bell, however, took a pass from Pangos and made a 3-pointer. Two Beltran free throws got the Jaguars to 59-58 only to have Pangos connect for his final 3-pointer.
Kelly Olynyk, who finished with 21 points and 10 rebounds, and Pangos scored 17 of Gonzaga's last 21 points. Pangos, who was 4-of-9 from 3-point range, ended with 16 points.
Beltran paced Southern with 21 points.