Wales huffed and puffed for parts of a 23-15 success, and they could not build on a 14-point lead interval lead secured by first-half tries from wing Alex Cuthbert and centre Scott Williams.
Italy fought back through a second-half try double by their impressive centre Michele Campagnaro, but Wales still made it 10 wins from their last 11 Six Nations starts.
"It was tough," Wales head coach Gatland said. "We've got to give Italy a lot of credit.
"We are pretty satisfied. There were some real positives, and there were some things to work on.
"We definitely kept them in the game in terms of a couple of crucial turnovers and an intercept, but we are off to a reasonable start and I wouldn't be surprised if Italy win a few games in this competition.
"It will focus the minds for next week (when Wales tackle Ireland in Dublin). It wasn't an easy run-out and it was tough, which was the way we wanted it.
"There are just are a few things for us to fix up on in terms of making sure we are a bit more accurate and clinical. We know we get stronger as a tournament goes on. It's that first one, and now we can look forward to the rest of it."
Gatland had short shrift for those who might raise last year's Brian O'Driscoll saga on the British and Irish Lions tour building towards next Saturday's Dublin showdown.
Gatland, in his role as Lions coach, received scathing criticism in many quarters after he dropped O'Driscoll for the deciding final Test against Australia. The Lions won 41-16.
"As far as I am concerned, that's dead and gone," Gatland added.
"It's not about Brian and myself. Brian has moved on, I have moved on. It is about 46 players and two nations playing an international next weekend.
"The game is more important than myself and Brian, and if people want to make an issue out of that for next week, to me it is just lazy journalism. Move on.
"It (the Ireland game) is a massive challenge for us. It is going to be a huge game for both teams. If Ireland manage to beat Scotland (on Sunday), you've got two teams coming together from opening victories on the first weekend.
"It will be interesting to see how they go (against Scotland) tomorrow."
On Saturday's performance, Wales captain Alun-Wyn Jones said: "It probably looked as if we were under more pressure than we actually felt out there.
"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 today."