Anscombe's under-strength side men were outscored by four tries to two during an out-of-sorts display, but a dominant performance from their pack led to a late penalty try converted by the reliable boot of Niall O'Connor.
And New Zealander Anscombe admitted his side, shorn of several players on international duty, had been fortunate to escape with victory.
"We have used our get out of jail card," he said.
"Let us be honest, we all know that. But we won. We are smiling at the fact we have four points and are now six points clear at the top of the table.
"We were flat and we just have to ask how we can improve rather than just forgetting about it.
"We have five days now to prepare for the Treviso game and we know they will be harder again. We have to get ourselves up for that game and fix what went wrong in Italy."
The visitors were heavily reliant on O'Connor, who kept his side in touch throughout.
Halfbacks Tito Tebaldi and Dan Halangahu both scored tries for the Italians, with Andries Van Schalkwyk scoring twice.
Paddy Wallace scored one of his own for the Irish Province, with O'Connor kicking five penalties.
Though admitting his side had under-performed, Anscombe did reflect on what he deemed had been poor preparation for the encounter.
"I don't want to make excuses, but we spent 14 hours getting there," he said.
"I think we have to be better as an organisation about how we organise our travel.
"Getting to a hotel at 11 o'clock on the night before a match is not ideal preparation."
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