But Ulster were too strong and they powered to victory thanks to Andrew Trimble, Ian Whitten and Darren Cave tries - with Niall O'Connor adding 14 points with his boot.
Despite collecting just one point in their opening three games, Moffat believes their first win is just around the corner.
"We gave away a couple of sloppy tries, it was a strange game and there were a lot of penalties in it for both sides," stated Moffat.
"We couldn't get any momentum or rhythm but I thought we looked dangerous at times when we had the ball and when we did have the ball, I thought we could score at any time.
"There was a good number of basic errors by us like at the restarts and two or three set pieces which meant we couldn't give ourselves a decent platform.
"We put great effort into the game considering the mount of ball Ulster had and I thought in a perverse sort of way we defended well but gave away a couple of soft tries.
"A lot of the time we did defend well but it was just a couple of lapses and every time we looked like getting into the game, Chris Paterson missed a penalty which he normally wouldn't miss, or we'd give a penalty away.
"It's not all doom and gloom and there is a thin line between winning and losing at this level. We have got to not lose belief in ourselves because we can play good rugby.
"We just need to get rhythm and momentum into our game, not lose our belief and we'll get there," he added.