Partick's top scorer Juan Ramon Escalas had already wasted a free-kick by the time Thistle went ahead in the ninth minute when Scotland goalkeeper Craig Gordon pushed an inswinging corner onto the bar and Armand One reacted quickest to head in the rebound.
Despite repeated attempts from midfielder Phil Stamp to link up with Kevin McKenna, Hearts were never able to pressurise the first division side and Escalas should have made more of a counter-attack which pitted three forwards against three defenders.
Paul Hartley found the Partick net midway through the first half after beating keeper Kenny Arthur to the ball on the edge of the penalty area but Hearts' top scorer was clearly offside.
Wyness also had a goal disallowed when the officials ruled that the ball had gone out of play before Ramon Pereira crossed for Wyness to tap home.
It was third time lucky for the Edinburgh club in the 34th minute when Stamp's corner caused confusion and was toe-poked into the net by midfielder Neil MacFarlane.
Hearts were finally dominating the game and Wyness and Pereira both went close before half-time. Hartley was outstanding after the break with a series of dazzling runs and a free-kick that was fractionally off target but One kept his team in the hunt with a vicious right-foot shot which was tipped over by Gordon.
A shot and a header were both saved by Arthur and then with time running out and an extra half-hour looking certain, Wyness pounced on a lucky deflection six yards out to send Hearts through.