Thompson's free throw with 3:10 remaining put the Broncs (11-9, 6-2 Metro Atlantic Athletic Conference) ahead to stay, 60-59.
Tavon Sledge's layup for Iona had tied the game 59-59, ending a stretch of 5:14 without a field goal for the Gaels (12-7, 6-2).
Iona's total matched its season low. The Gaels, who had won six straight, were averaging 82.6, third highest in Division I.
Pereira made his second straight start in place of Nurideen Lindsey, who is recovering from a concussion, and hit two 3-pointers 14 seconds apart, when Rider erased a 25-14 first-half deficit to lead 35-30 at halftime.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.