Cove took the lead in the 52nd minute when McAllister headed in Logan's cross, but Steven Boyd equalised. McAllister, though, tapped in before wrapping things up from the penalty spot with a minute left.
Falkirk sit third after coming from behind to beat bottom club East Fife 2-1. Second-half goals from Michael Ruth and substitute Aidan Keena turned the game after Kieran Millar’s first-half opener.
Airdrie are fourth after their 2-1 win over Clyde. Callum Smith put the hosts in front early on, David Goodwillie hit the equaliser but Calum Gallagher's strike just before the hour secured all three points.
Dumbarton dropped to fifth after an unexpected 5-0 loss at strugglers Peterhead.
Second-placed Queen's Park host Montrose on Sunday.