Defeats for both teams in the opening round of the European tournament meant this one was all about survival, the two points and staying in contention for a lucrative last eight place.
For Ulster the end is nigh; for Swansea there is still hope. But they will have to pick their game up considerably over the next four group fixtures if they are not to go the way of their visitors.
This was a drab affair, littered with errors and largely dominated by the boot, Thomas kicking seven penalties for Swansea and opposite number Stuart Laing three for Ulster. Thomas finished with an admirable return of eight successes from ten attempts.
He started punishing Ulster with a penalty as early as the third minute, after he was pulled back while chasing his own chip kick. Thomas then converted the centre Scott Gibbs's 18th-minute corner try.
But any thoughts that the Lions' hero's score would provide the release of tension for Swansea to open out never materialised, the Ulster centres Maurice Field and James Topping, along with back row men Keith Gallick and Andrew Ward, epitomising a never-say-die spirit.
But as a contest it never rose above the distinctly mundane and perhaps the whole affair was best illustrated by the bizarre comings and goings surrounding the replacement of international lock Paul Arnold.
His original replacement, Danny Niblo, was replaced himself by two temporary replacements who themselves both went off before Niblo rejoined the action at half time.
Confused? Referee Ken McCartney almost certainly was and just to add another twist the Scottish official almost had to go off himself after being bowled over and hurting his neck in one of Ulster's better attacks.
But the bottom line was that Ulster simply did not have the ammunition to pull off an upset - even if the All Whites were decidedly off colour and repeatedly kept the ball close rather than testing the visitors out wide.
At least Ulster provided a steady stream of penalties that Thomas, despite the gusting wind, turned into valuable points. "It was pretty windy out there and it was always going to be difficult with the kicking," Thomas said. "But I was just happy to see the majority of them going over. I guess it could have gone either way and everyone at Swansea knows we have a lot of work to do."
A fair amount of that work should involve the tackle bags, as Swansea miss too many tackles for comfort. But at least they sent the home section of the meagre crowd home happy as they finished with a flourish. Hugh Thomas, on for the centre David Weatherley, scorched away for a superb try that sealed victory.
Swansea: L Evans; R Rees, D Weatherley (H Thomas, 12), S Gibbs, A Harris (A Williams, 63); A Thomas, R Jones; C Loader, G Jenkins (capt; C Wells, 77), C Anthony, T Maullin, P Arnold (D Niblo, 25), R Appleyard, D Thomas (A Reynolds, 56), P Moriarty.
Ulster: R Morrow; S Coulter (A Park, 53), J Topping, M Field, J Cunningham; S Laing, S Bell; R Mackey, S Ritchie (M Patton, 68), G Leslie, T McWhirter, G Longwell, S McKinty (capt), A Ward, K Gallick.
Referee: K McCartney (Scotland).Reuse content