The teams waited out a two-hour rain delay before the Storm Chasers scored unearned runs in the first and second innings for a 2-0 lead.
The Storm Chasers increased their lead over New Orleans to 4-0 in the fourth thanks to an RBI single by Anthony Seratelli and a run-scoring double by Wil Myers.
Max Ramirez's solo home run in the fifth inning extended Omaha's lead to 5-0.
All of the runs were charged to New Orleans starter Elih Villanueva (9-8), who allowed nine hits in 4 1/3 innings to take the loss.
Adcock (8-5) limited the Z's to one run and six hits over seven innings, earning the win. His wild pitch with two outs in the sixth permitted the Zephyrs' lone run to score.
New Orleans dropped to four games behind first-place Albuquerque, who won their game in Memphis and lowered their magic number to clinch the division to six games.
The Zephyrs and Storm Chasers will play the finale of the five-game set on Sunday at 2:05 p.m. Right-hander Tom Koehler takes the mound for New Orleans, opposing Omaha left-hander Chris Dwyer.