Early penalties from Paul Warwick had given Connacht a 6-0 lead but the strong Ulster pack soon took command - and Kevin Maggs, on his Celtic League debut, showed what a useful signing he will be as he constantly troubled the home defence, nearly getting the game's first try after 20 minutes.
It was only seven minutes later that David Humphreys narrowed the gap with his first penalty after Matt Mostyn had been sin binned.
Continued Ulster pressure soon found a crack in the Connacht defence and when John Muldoon was yellow carded the extra man advantage allowed Bryn Cunningham to find a gap and feed Neil Doak who set up Bowe for the opening try.
Ulster took that 8-6 advantage into the break and Bell extended the lead when he brushed through two or three tackles to score in the corner.
Humphreys added two penalties but Connacht continued to battle and were rewarded with a late try from Conor O'Loughlin which secured the losing bonus point.
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