The Wasps shocked their top-flight opponents when Robert Thomson opened the scoring in the first period.
However, Livi, who also saw a Jay Emmanuel-Thomas goal chalked off, turned the game on its head during a devastating four-minute spell after the break.
Alan Forrest restored parity before Mullin, a deadline-day capture from Ross County, secured the points with a somewhat fortuitous looping effort.
The result cements Livi's place at the top of Betfred Cup Group H with two wins from two but it was far from routine.
Alloa broke the deadlock after 30 minutes when a deflected Liam Dick shot was superbly saved by Max Stryjek, only for Thomson to react quickest and prod home the loose ball.
Livi did finally breach the Alloa defence just 30 seconds into the second period, with Forrest slotting into an empty net after Parry could only palm an Emmanuel-Thomas drive into his path.
And a quick-fire comeback was completed four minutes later when Mullin's deep cross deceived everyone and drifted into the net.
Livingston almost added gloss to the scoreline when Forrest rattled the cross-bar after turning Dick and Edin Lynch inside-out in the box.