Danny Grainger had given Hearts the lead just before half time, but Marc McNulty equalised with a blistering strike just short of the hour mark.
Home skipper Marius Zaliukas made amends for his error in the build-up to Livi’s leveller by grabbing two goals in two minutes to secure his side’s passage to the next round.
Hughes said: “I felt our shape and tactically we were good, and I’ve no criticisms but we are a better passing side than we were tonight. I’m disappointed with the way we conceded the goals, but that’s what you get when you’re working with young kids – two set pieces and a deflected goal. In saying that, Andy McNeil pulled off some wonderful saves too.
“We gave them a fright but unfortunately we couldn’t hang on to it for long enough.
“I think we had the best player on the park in Stefan Scougall. He was outstanding and really showed his class.”
Hearts took the lead on 41 minutes when a Grainger shot from 20 yards spun wildly off a stray leg and past McNeil.
But on 57 minutes, Livi equalised when McNulty robbed Zaliukas on the edge of the box and smashed home in front of a vocal travelling support.
Hearts’ quality shone through in the closing stages, however, as Zaliukas responded twice from corners to put the tie beyond the Lions.