Italy threatened to spring an upset when the second of Michele Campagnaro's two tries saw them close to within a single score, though, before a third penalty from Leigh Halfpenny ended the resistance with six minutes remaining.
Head coach Gatland admitted that it was a nervier ending than he would have liked but Jones says the players on the pitch were always confident they would close out the game and get their quest for a third successive tournament title up and running.
"It probably looked as though we were under more pressure than it felt on the park. One of their tries was an interception and one was from a kick. It was a case of locking down the hatches and sticking to what we'd been doing," said Jones, who captained the side in place of Sam Warburton, just as he did when the Lions secured series glory in Sydney in the summer.
"The message was to be simple on our exits, simple in our play and to get back into our patterns. With the kickers we've got, we knew that if we got opportunities for points we'd just take the three and keep stretching the scoreboard, which we did.
"It's a big potential banana skin to get out of the way. We were acutely aware of the performances Italy put in last year and we realised how difficult it was going to be."
Wales attempted to keep the ball alive at almost every opportunity under a closed roof at the Millennium Stadium, with Williams' 38th-minute score a prime example after a wayward lineout didn't prevent the ball being shipped along the line for Jamie Roberts to cut through on halfway.
That approach brought a few more handling errors than Jones wanted but the Ospreys second row says that is perhaps to be expected, even though he is seeking improvement against Ireland in Dublin next weekend.
"A willingness to play with ambition comes with mistakes," added Jones. "We were looking to play and use the ball - something we've been criticised about in the past.
"We seemed to spend a lot of time around the fringes of their 22 but we probably squandered it and were a bit soft. Ultimately, we started to come back into the game but we couldn't keep the tempo up, particularly at set-piece time for certain reasons. But we need to hold on to the ball when we get into those (scoring) areas.
"In the past we've seemed to get more points the more phases we go through and we really don't want to put ourselves under pressure at those times."