The Blues fought back on two occasions against the Irish region having fallen behind 16-3 after 28 minutes and then 24-11 after Robin Copeland's yellow card.
However tries from Andrew Trimble and Dan Tuohy, his second of the night, thereafter handed Ulster a 39-21 bonus-point victory at Ravenhill.
Luke Marshall and Jared Payne also went over for Ulster with Paddy Jackson slotting three penalties and Ruan Pienaar one as well as a conversion.
Leigh Halfpenny recorded 11 points with the boot while Copeland, prior to his sin binning, and Richard Smith crossed for tries for the Blues.
And, while director of rugby Davies was pleased with the spirit among his players, he admitted a lack of precision cost them dearly at Ravenhill.
"It's disappointing to go and score 21 points and not take anything back with us," he said. "It's not an easy place to come; it's a new fantastic stadium with a great atmosphere.
"It was about us going and fronting up and producing a performance full of endeavour, which we did.
"But we weren't accurate enough to go away with anything, which is disappointing when you think of the effort, commitment and physicality we put in.
"Our driving game went well. The pack went well and after we had a few creaky scrums but sorted that out.
"We drove and got a good score, we also showed character when down to 14 men after Robin was yellow carded.
"The players showed a lot of character to play from deep at the end, but it wasn't enough against a strong defence and they got a couple of tries late on which lop-sided the score line a little."