He struck out five and walked four after taking over for starter Justin Bristow in the fifth inning.
East Carolina scored three runs in the sixth to rally from a 3-1 deficit, keyed by Corey Kemp's 16th homer and RBI singles from Dustin Harrington and Harrison Eldridge.
The fifth-seeded Pirates (40-18) will face the UAB-Houston loser in another elimination game Friday.
Rice (42-13) entered ranked No. 5 nationally by both Baseball America and Collegiate Baseball, but went 0-2 in a league tournament for the first time since 1995.