There was huge relief at the rain-lashed Sportsground as Dan Parks' late drop-goal guided Connacht to a long-awaited second win in the RaboDirect PRO12.
Pat Lam's men had endured eight successive defeats since beating Zebre in round one back in September, falling to the bottom rung of the league table.
But despite going 8-0 down to the Dragons inside 16 minutes, Connacht - helped by a dominant scrum and line-out - fought their way to a gutsy three-point win.
Daniel Evans' early unconverted try was the difference between the sides at the break, the wind-backed Dragons establishing a 8-3 interval lead with Jason Tovey and Parks swapping penalties.
Two more successful kicks from Parks and Tovey followed and Lyn Jones' side maintained their 11-8 lead despite losing prop Owen Evans to the sin-bin.
But Connacht produced the stronger finish with Parks setting up replacement Fionn Carr for a 75th-minute try and then nailing his vital drop goal with just over a minute left.
The sixth-minute withdrawal of centre David McSharry, who picked up an ankle injury, forced Connacht to shuffle their back-line with Carr introduced and both Robbie Henshaw and Gavin Duffy making positional switches.
It was Henshaw's missed tackle that allowed Ross Wardle to break through midfield in the eighth minute and tee up full-back Evans for his five-pointer.
Tovey, who had missed an earlier penalty, was back on target with a 16th minute penalty as the Welsh region kept pressing with a strong wind at their backs.
However, Connacht then used their solid scrum to build some momentum, with youngsters Eoin McKeon and Michael Kearney both prominent in the set-pieces.
A scrum penalty allowed Parks to knock over a 33rd minute penalty and as the strength of the wind dropped noticeably, Connacht's forwards exerted more influence with the visitors' chances to attack curtailed.
The gap was down to just two points after Parks kicked a penalty early in the second half, punishing young lock Cory Hill for being caught offside.
A good spell of possession saw the Dragons hit back with Tovey left-footing a long range penalty, but a yellow card was looming for the visitors as their own penalty count increased.
Loosehead prop Owen Evans was the man to see yellow on the hour mark, referee Alain Rolland sin-binning him for a foot trip. However, the home side suffered further frustration when Parks failed to convert the resulting penalty.
The Dragons kept it simple during a weather-beaten final quarter, with number eight Toby Faletau leading by example in terms of his sharpness on the ball.
But Connacht - driven on by man-of-the-match John Muldoon - pounced for a levelling try with five minutes left, Parks' deft crossfield kick being dotted down by the unmarked Carr out wide.
The former Scotland international narrowly missed his conversion attempt to the left, however, and Connacht needed to work their way downfield to set up a second shot at the posts for him - and he delivered.