A late try from Fionn Carr and a drop-goal from Dan Parks handed Connacht a hard-fought 14-11 victory over the Dragons in Galway.
Dragons took the lead through a Dan Evans try before Jason Tovey added a penalty, with Dan Parks scoring a penalty in reply to leave the hosts five points adrift at half-time.
Connacht started well in the second half and scored an early penalty through Parks, but saw it cancelled out by Tovey with one of his own shortly after.
Connacht were dealt an early blow when Dave McSharry was forced off with an injury and conceded a penalty a minute later, but were let off the hook as Tovey's kick drifted wide.
The visitors made up for the miss shortly after as they broke down the right wing before Dan Evans went over to open the scoring. Tovey missed the conversion but was successful on his third attempt with the boot after the Dragons were awarded another penalty, giving the visitors an 8-0 lead.
The wind was causing problems for both sides, and Parks failed to find touch with the penalty as the hosts looked to get a foothold in the game.
Connacht had to wait a while for their first real opportunity to test the Dragons' line, but when it finally came, Kieran Marmion's pass was intercepted, and only a Matt Healy tackle prevented the visitors from scoring their second try of the game.
Connacht did find a way back into the game when they were awarded a penalty from the scrum in Dragons half. Parks opted to kick to goal and successfully navigated the testing conditions to reduce the deficit to five.
The hosts looked to be gaining momentum and were awarded another penalty after Mick Kearney stole the ball from a Dragons line-out, but failed to make it count and Connacht went in at half time 8-3 down.
Connacht started the second half brightly and, with the wind on their side, scored their second penalty of the game through Parks.
The Dragons responded well to having their lead cut, though, and Tovey kicked his second penalty of the game after the hosts were penalised for not rolling away.
Indiscipline finally cost Newport when they conceded their 11th penalty of the match, as Owen Evans was sent to the sin-bin, but the hosts missed an opportunity to go within two as Parks' penalty went wide.
But with just five minutes to go, Connacht equalised as Carr scored in the corner. The conversion to take the lead, though, was missed by Parks.
With the home side in the ascendancy, the fly-half was presented with an easier opportunity to win the game when he positioned himself for the drop-goal. The kick sailed between the uprights to put the hosts 14-11 up in the dying moments, leaving the Dragons with no time to respond and sealing the victory.
Pens: Parks 2
Try: DJ Evans
Pens: Tovey 2
Yellow Card: O Evans
Connacht: 15 Robbie Henshaw, 14 Gavin Duffy, 13 Eoin Griffin, 12 Dave McSharry, 11 Matt Healy, 10 Dan Parks, 9 Kieran Marmion, 8 Eoin McKeon, 7 Jake Heenan, 6 John Muldoon, 5 Craig Clarke (c), 4 Mick Kearney, 3 Nathan White, 2 Jason Harris-Wright, 1 Brett Wilkinson.
Replacements: 16 Dave Heffernan, 17 Denis Buckley, 18 Rodney Ah You, 19 Andrew Browne, 20 George Naoupu, 21 Paul O'Donohoe, 22 Jack Carty, 23 Fionn Carr.
Dragons: 15 Daniel Evans, 14 Will Harries, 13 Pat Leach, 12 Ross Wardle, 11 Hallam Amos,10 Jason Tovey, 9 Richie Rees, 8 Toby Faletau, 7 Nic Cudd, 6 Lewis Evans, 5 Cory Hill, 4 Andrew Coombs (c), 3 Francisco Chaparro, 2 Sam Parry, 1 Owen Evans.
Replacements: 16 T Rhys Thomas, 17 Hugh Gustafson, 18 Dan Way, 19 Matthew Screech, 20 Ieuan Jones, 21 Wayne Evans, 22 Steffan Jones, 23 Ashley Smith.
Referee: Alain Rolland (Ire)