Dan Parks opened the scoring for the visitors but from there the Dragons dominated the scoreboard to end their dismal run of defeats.
Tom Prydie was on target either side of Pat Leach’s crucial try in the closing stages of the first-half
Edwards’ men had crashed to seven consecutive losses and were in desperate need of a victory to kick-start their season.
Former Cardiff Blues fly-half Parks booted Connacht into an early lead but both sides were failing to deal with the difficult conditions.
Tom Prydie hit-back as Connacht saw both Jason Harris-Wright and Andrew Browne sent to the sin-bin in quick succession..
The Dragons laid siege to the Connacht twenty-two and the pressure eventually told as Leach slid over.
Prydie was agonisingly off-target with the touchline conversion but the try propelled the Dragons into an 8-3 half-time lead.
They made a quick start to the second half and immediately put Connacht under pressure. Prydie fired over a pair of penalties to ease the Dragons into a 14-3 advantage.
Connacht rallied at the death but the Dragons coped with everything thrown at them as they hung on for a long-awaited victory.
