But the Firhill outfit allowed the visitors back into the game and, inspired by 14 points from the boot of Ian Keatley, Connacht broke their away hoodoo - much to the disgust of Lineen.
"I am incredibly frustrated and disappointed," said Lineen.
"We certainly put the effort in but we played dumb rugby at times.
"We started well but stopped going forward. Connacht had a game plan they stuck to - and we let them do it."
Ulster made sure it was a miserable night for the Scottish sides by downing Edinburgh 29-21 at Ravenhill, securing top spot in the process.
Chris Paterson kicked eight points for the Gunners while fellow Scotland international Ben Cairns and Dutch winger Tim Visser crossed for tries.
However, 14 points from the boot of fly-half Niall O'Connor and scores from Andrew Trimble, Ian Whitten and Darren Cave secured Ulster the points and sent them top of the Magners League pile.