The Dragons head to the Galway Sportsground on Saturday knowing a loss to Connacht will knock them out of contention for the Celtic League title.
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The Gwent region are tucked in behind the two front-runners, but manager Jim McCreedy knows that any slip-up will turn the title race into a two-horse affair between Ulster and Llanelli Scarlets.
"Connacht have a tremendous record this year," McCreedy said. "They've beaten quite a few quality sides and they're probably the surprise package of the season along with ourselves.
"They'll be looking to bounce back after their unfortunate defeat in the Parker Pen series against Harlequins at the weekend and they'll be keen to beat a side above them in the table."
The Dragons have won 14 of their 19 games this season - one more than Ulster and the Scarlets - but a lack of bonus points has consigned them to third in the table.
"We have missed out on the bonus points in a couple of games, and that may come back to haunt us in the final reckoning," McCreedy admitted.
"But what's gone has gone and we'll continue to play the way we've been playing. When you go to a place like Connacht, it's hard enough to win the game, let alone talk about the bonus point. In games like this, the bonus point is literally a bonus, so we'll just concentrate on winning.
"You have to say Llanelli are everyone's favourites to win the title at the moment. They have two home games left, while we have two away games, but everyone at The Dragons is considering the season a success.
"Whether we're still in the running in the last weekend or whether we finish second or third, we're proud of what we've achieved.
"At the beginning of the season, we were everybody's favourites to finish last of the Welsh regions and be relegated to the Parker Pen tournament, but we've reached our aim of being in the Heineken Cup next season with three games to spare."
McCreedy said Dragons coach Mike Ruddock had a relatively injury-free squad to select from for the Connacht match, although hooker Paul Young (knee) and Scotland backrower Ross Beattie (thigh) will miss the match. The club is hopeful both forwards will be back for the last two games of the tournament.
Connacht: M Mostyn, T Robinson, D Yapp, S Moore, C McPhillips, E Elwood, C O'Loughlin, A Clarke, J Fogarty, P Bracken, D Browne, A Farley, P Neville, T Carter, M Carroll.
Replacements: B Jackman, D McFarland, M McCarthy, J O'Sullivan, M Walls, D Hewitt, M McHugh.