Allen, who was 0 for 8 in the CWS when he came to bat in the third inning, went 3 for 5 with three RBIs.
Josh Laxer (3-2) worked 22⁄3 innings of relief for the win, and Aaron Greenwood went the last 11⁄3 innings for his fifth save. Jordan Kipper (8-3), who relieved struggling TCU starter Tyler Alexander, took the loss.
Allen had a two-out, two-run double in the second for a 3-0 lead, the tie-breaking base hit in the seventh, and he scored an insurance run after he singled in the ninth.
Alexander, TCU's first 10-game winner in four years, struggled for a second straight start after he won six in a row. He gave up consecutive hits to start the fourth and left with his team down three runs.
Ole Miss starter Sam Smith lost his control and didn't make it out of the fourth, either. Laxer came on with the bases loaded and issued a two-out walk to Cody Jones that tied it 3-all. That marked the first time an inherited runner scored against the Rebels' bullpen in nine NCAA tournament games.
Each team added a run in the fifth, with Cron driving Laxer's high fastball 10 rows into the stands in left field for the CWS' second homer in 10 games.
Laxer and freshman Wyatt Short combined to retire nine straight after that. TCU threatened in the eighth, putting runners on first and second with one out. But Short struck out Dylan Fitzgerald, and Greenwood came on and got Keaton Jones to ground to shortstop, with Errol Robinson pumping his fist after throwing out Jones.
Mississippi 6, TCU 4
• Vanderbilt vs. Texas,1 p.m., ESPNU