The Galway crowd had little to cheer in the opening 40 minutes as a Jason Tovey penalty and a converted Richard Fussell gave the Dragons the edge at the interval – 10-3.
But Michael Bradley’s half-time words had the desired effect for the hosts as they raised their game after the break to score three tries in the space of 11 minutes.
After that, the belief spread right throughout the Connacht side and further efforts from Niva Ta’auso and Fionn Carr secured a much-needed five points for Bradley’s men, ahead of their European Challenge Cup quarter-final trip to Northampton next week.
The Dragons have only won once in Ireland since October 2004 but they made a promising start with Wales Under-20 starlet Tovey kicking a seventh-minute penalty.
His opposite number Ian Keatley knocked over a drop goal in reply but neither side was able to stamp their authority on proceedings.
The Dragons, with Tom Willis, Luke Charteris and Joe Bearman standing out, looked the more cohesive unit and they gained a decent bit of momentum before speedster Fussell thundered through for his seventh try of the campaign.
Racing to the top of the try-scoring charts, Fussell got the better of three defenders before crossing close to the posts, for Tovey to convert.
Connacht captain John Muldoon and hooker Sean Cronin both went close to scoring tries at the other end, but Paul Turner’s charges had done enough to warrant a seven-point half-time buffer.
However, the second half produced a complete turnaround. Players like Conor O’Loughlin, the fit-again Ta’auso, out-half Keatley and the heroic Muldoon led from the front as Connacht ended a run of five straight league defeats.
Ta’auso produced a brilliant offload for Gavin Duffy to hare through a gap and put the supporting Ray Ofisa over for a 44th-minute try. Keatley’s conversion levelled the game.
With O’Loughlin’s swift passing coming to the fore, two more breaks from the influential Duffy and some forward grunt then set up a try for winger Carr in the left corner.
Keatley missed the conversion but he regained his poise to add the extras to a subsequent try in the same corner, with Australian Liam Bibo the scorer after the Dragons rearguard was carved open again.
With their confidence rocketing and memories of last week’s heavy defeat in Edinburgh banished, Connacht pressed on with the bonus point within sight.
Their back row destroyed the Dragons, aided by Johnny O’Connor’s return, hoovering up a huge amount of ball and Connacht also benefited from a very solid scrum and lineout.
The Dragons looked dangerous with ball in hand – Fussell and Aled Brew particularly – but the hits were coming in from Connacht and one huge tackle by Muldoon set up a successful penalty for Keatley.
Ta’auso showed excellent pace to intercept a pass on his 22 and charge through for the bonus point score before Carr took his league tally to five tries after a pinpoint Keith Matthews pass had unlocked the defence.
The damage was done and Lydiate’s late try, converted by replacement Shaun Connor, was only for consolation as Connacht celebrated their first win over a Welsh side this season.