The visitor is bottom of the Six Nations standings without a point after three matches, with Gatland now taking the opportunity to shuffle his pack for its visit.
The bonus point win pushes Wales above England into second place in the Championship table.
Second row Hill went over from close range to get his first global try, before Anscombe maintained his ideal kicking display to make it 24-7.
"But that's probably to be expected when guys have not played together".
It was not all rosy for Wales, though, and they would be reduced to 13 men for a few minutes after Gareth Davies was sent to the sin bin for a deliberate knock on. But we achieved what we wanted to do and we have our destiny in our own hands next week.
Liam Williams saw yellow for a risky tackle on his opposite number Matteo Minozzi at the end of the first half and did not return to the field, Gatland sending on Leigh Halfpenny when the sin-bin period was up. "We had no problems with the yellow card", Gatland said. "I was disappointed. Whether he had to make the tackle, did he need to go that high?"More news: Meghan Markle joins Queen Elizabeth at Commonwealth Day service
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"The thing with Liam is he's such an emotional player". "He knows if he isn't comfortable in that environment, we will support him and the union (Welsh Rugby Union) will support him 100 per cent for whatever happens with him".
Gatland said replacing Williams "was about sending a subtle message really".
Those contentious decisions were no matter, however, as the Azzurri were dealt their 16th straight defeat in the competition.
Seven of the 12 victories they have accumulated in their 19 seasons in the tournament have come against Scotland, whose revival in the last couple of seasons has left the wooden spoon barely contested.
Italy head coach O'Shea has made just one change to his starting XV for the Cardiff clash, naming Giulio Bisegni in his midfield for the first time in over a year.
"It's just about working on what we've been doing in training and express yourself".
The former Ireland fullback added: "If you go around the clubs, see the Under-20s beating Wales, and look at the players coming through, we are building a team and we are building a system piece by piece".