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The two-time defending champions got off to the perfect start with last year's Championship top try scorer Alex Cuthbert opening his account after just four minutes.
Centre Scott Williams also crossed before the interval to give Wales, who are chasing an historic third consecutive RBS 6 Nations crown, a 17-3 lead at half-time.
Back came Italy after the break however and 20-year-old centre Michele Campagnaro crossed twice, pulling the Azzurri back to within five points with 11 minutes remaining.
Leigh Halfpenny's 73rd-minute penalty gave Wales an eight-point cushion and Gatland's side saw out the victory but the head coach admitted improvements must be made before Saturday's trip to face Ireland in Dublin.
But Jones insisted Wales were in control despite Italy's second-half rally.
"It probably looked as if we were under more pressure than we actually felt out there," said Jones, captaining the side with Sam Warburton on the bench.
"We were a bit ragged at times, but we showed ambition and a will to play.
"We showed ambition, and at times you get mistakes, but that was a big potential banana skin, if you like, to get out of the way.
"We were acutely aware of the performances Italy put in last year, hence the realisation of how difficult this was going to be."