The new signing from Middlesbrough was joined on the scoresheet by Matt Taylor and Ross Wallace as Sean Dyche's men saw off their Serie A opponents at Turf Moor
With just over a week to go before Chelsea head to Turf Moor for Burnley's Premier League opener, Sean Dyche's men came from behind after the visitors took the lead courtesy of a Lazaros Christodoulopoulos header.
But pre-season signings Matt Taylor and Jutkiewicz were both on target to change the complexion of the game, with the latter making it six goals in as many appearances since joining from Middlesbrough.
Substitute Ross Wallace produced a moment to magic to add a third with a 25-yard free-kick eight minutes from time and seal a home win that will boost confidence at the Lancashire club.
Verona, who finished a credible 10th on their Serie A return in 2013-14, created a flurry of early chances, with Rafael Marquez's shot deflected wide before a Luca Toni effort also missed the target from the resulting corner.
Jutkiewicz directed a 20th-minute header over before Danny Ings curled a shot just wide from the edge of the penalty area as Burnley grew in stature.
The game burst into life with two goals within the opening two minutes of the second half.
Christodoulopoulos first headed home Toni's lofted pass before the hosts responded immediately, Kieran Trippier, Ings and Taylor combining for the former Portsmouth and Bolton Wanderers midfielder to net from 10 yards.
And the Burnley comeback was complete when Jutkiewicz continued his purple patch in front of goal after 61 minutes.
Ben Mee delivered an exquisite left-wing cross and Jutkiewicz met it with an unstoppable header before being withdrawn, presumably as a precautionary measure to keep him fresh for the upcoming season opener with Chelsea.
Ings created and spurned a couple of chances before Wallace curled home a stunning late set-piece to make it 3-1 after Dean Marney had been bundled over by Artur Ionita.