Ulster were trailing 7-3 at the break with Ospreys fly-half Matthew Morgan pulling the strings and flanker Sam Lewis touching down after just ten minutes.
The Irish province responded however and after a Ruan Pienaar penalty in the first half, Cave's score - converted by the South Africa scrum-half - put Ulster in the ascendancy.
They held on for the victory and as a result, they leapfrogged the Ospreys in the RaboDirect PRO12 table and moved into second place.
But Anscombe, who signed a contract extension earlier this week to remain at Ravenhill until June 2015, was far from impressed with his side's performance.
"We missed more tackles than we did in combined in every game since I have been here. Our defence and kicking was poor," he said.
"It was a real first game back from holiday and it looked like it."
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