In a remarkable game, the league leaders were outscored four fries to two by the Italian whipping boys, and needed a late penalty try to eventually get out of jail.
The absence of their Irish contingent notwithstanding, Ulster were nevertheless out of sorts, and bore little resemblance to the side which has beaten all comers this season.
Their inertia, however, seemed reasonable once Anscombe threw light on their preparation.
“I don’t want to make excuses, but yesterday we spent fourteen hours getting here,” said the relieved New Zealander.
“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.”
Flagging from the word go, Ulster relied heavily on the boot of Neil O’Connor to keep them in touch on a day when Zebre – still chasing their first ever win – went try-crazy.
They scored four in all, one each for halfbacks Tito Tebaldi and Dan Halangahu complimenting a brace by Andries Van Schalkwyk.
Sandwiched in between, Paddy Wallace scored one of his own for Ulster, but the writing still appeared to be on the wall as the game entered its dying embers.
Cue two collapsed scrums by the Zebre in front of their posts, and a timely intervention by referee Neil Paterson.
Zebre scorers: Tries: Tebaldi, Halangahu, van Schalkwyk 2. Cons: Halangahu. Pens: Halangahu.
ZEBRE: Odiete, Sarto, Pratichetti, Pace, Castagnoli, Halangahu, Tebaldi, Aguero, Manici, Redolfini, Sole, Cazzola, Ferrarini, Cristiano, van Schalkwyk.
Replacements: Perugini for Aguero (52), Tripodi for Redolfini (65), Cattina for Ferrarini (65).
Ulster scorers: Tries: Wallace, Penalty try. Cons: O’Connor. Pens: O’Connor 5.
ULSTER: Andrew, Allen, Payne, Wallace, Cochrane, O’Connor, Heaney, Black, Brady, Afoa, Diack, McComb, Williams, McComish, Wilson.
Replacements: Court for Black (55), Herring for Brady (52).
Referee: A McMenemy (SRFU).