Beziers snatched a late victory at the Sportsground, but Connacht won this Parker Pen Challenge Cup tie on aggregate and were rewarded with a second round tie against Pau in January.
The heroic performance in France last week was enough and, indeed, Connacht were unlucky to have narrowly lost this second leg deep into injury time.
While the defeat was frustrating, the real black mark on an impressive Connacht second half display was the sending off of reserve prop Fabian Boiroux, who subsequently was handed a nine week suspension for an alleged head-butt.
Still, while he can hardly quibble with the dismissal, he might just feel a little aggrieved with the length of suspension.
Jean-Jacques Crenca was sent off for a similar offence in last week's Heineken Cup defeat to Northampton and received a mere four week suspension.
Connacht coach Michael Bradley had no complaints about Boiroux's red card, although he did suggest the prop had been intimidated.
"I wouldn't even attempt to explain what goes on between front row players and Fabian said he reacted to something that had happened just beforehand. But at the end of the day, we cannot condone a reaction like that.
Otherwise, the Connacht performance had to be admired. Beziers took the risk of playing against the elements in the first half and it paid off.
Connacht, though dominating territorially, found it difficult to put daylight between themselves and their visitors and paid the price in the 32nd minute when Beziers rumbled over for a try from a tapped penalty 10 metres out - Yannick Nyanga touching down with Federico Todeschini converting.
Eric Elwood had kicked a 14th minute penalty and Mark McHugh successfully kicked another from near half way to stretch Connacht's overall advantage to 14 points.
But the Beziers try clearly rocked Connacht and when the rain and wind abated, the French grew in confidence. It hardly helped that Connacht were struggling as Beziers snaffled three home throws to keep the heat on.
Even the most passionate Connacht supporters feared the worst at the break, but Bradley's half time words clearly had the desired effect. "We spoke about the need to start the second half well and the absolute necessity to get some kind of score," he said.
That score came in the third minute and it boosted Connacht's overall lead back up to 12 points.
In the circumstances it was a score of remarkable poise, teamwork and handling between forwards and backs before Mark McHugh slid over for a try near the corner.
The impact on Beziers was profound. Connacht's line out work improved considerably in the second half and so too did their success rate.
While Connacht boxed clever, Beziers hardly learned from their mistakes. They did hit back with a penalty from Todeschini midway through the half but continued to earn the wrath of referee Paul Adams for technical offences at the ruck and maul.
The crucial moment came with Boiroux's dismissal, however, and while Todeschini failed with the resultant penalty attempt, he knocked over one in lost time.
Beziers celebrated wildly at the final whistle, bemusing the Galway fans until they learned that Beziers sneak through to the next round as "lucky losers", a back door system that affords vanquished teams another chance if they lose on aggregate but win one of the two games.
Jean Pierre Elissalde, the Beziers coach, was delighted with the outcome: "That penalty was vital for us because we are still in the competition."
But Bradley had even more reason to smile. "I think the overall result proved this is a good Connacht team. The lads had to show a lot of character over the last two weeks to win in France and to survive a moderate first half display at home.
"I think that the guys probably deserved to win this match but we're happy to get through. Pau play a different style of rugby but we see it as another challenge."
CONNACHT: M. McHugh, M. Mostyn, D. Yapp, T. Allnutt (captain), C. McPhillips, E. Elwood, C. O'Loughlin, D. McFarland, B. Jackman, A. Clarke, M. McCarthy, D. Browne, M. Swift, J. O'Sullivan, P. Neville.
Replacements. M. Walls for O'Loughlin (60), T. Robinson for Allnutt (64), F. Boiroux for A.J. Clarke (66).
BEZIERS: G. Arandiga, P. Escalle (captain), J. M. Aue, Y. Audrin, D. Vainqueur, F. Todeschini, P. H. Broncan, A. Tolofua, S. Bruno, P. Cardinali, M. Koita, S. Hall, Y. Nyanga, A. Petrache, M. Uanboyev.
Replacements. G. Shvelidze for Cardinali, J. F. Pedesseau for Tolofua, O. Llari for Uanbayev (all 55)
Referee: P. Adams (Wales)