Matt Gatens had 10 points and seven assists for the Hawkeyes (2-0), who blew a 12-point halftime lead but hit 14 of 15 free throws in the second half to avoid the upset.
Tucker buried a deep 3 with 1:47 left to put Iowa ahead 67-62. The Hawkeyes got the ball back after Devin Gibson misfired from beyond the arc, but Omar Johnson buried a 3 in transition after an Iowa turnover to make it 67-65 with 33.1 seconds left.
Gatens hit two free throws to push Iowa's lead back to four, and the Roadrunners turned it over off a missed Gibson free throw with 11 seconds left.
Texas-San Antonio (0-2) opened the second half with a 13-4 run to pull within 44-41. Iowa pushed it back to 50-41 before the Roadrunners responded with a 16-4 run, highlighted by a pair of Joey Shank 3s, to take a 57-54 lead with 7:48 left.
Johnson scored 17 to lead Texas-San Antonio. The teams combined for 35 turnovers.