All Connacht's tries came from forwards with prop Rodney Ah You and replacement lock Mike Swift adding to Muldoon's tries while fly-half Dan Parks kicked two conversions.
Tom Pyrdie went over for the Dragons only try with 10 minutes to go, adding to an early Jason Tovey penalty.
Dragons drew first blood when Tovey landed a penalty and then just missed with another from half-way.
Yet that was to be the first and last time the home side were in front as Connacht's pack hammered the Welshmen.
Locks Ally Muldowney and Mick Kearney took everything in the front and centre of the lineouts as did number eight Eion McKeon at the tail.
And Connacht's front row of Dennis Buckley, Jason Harris-Wright and Kiwi Ah You had the home trio in so much trouble that Dragons head coach Lyn Jones switched the entire unit after 28 minutes.
Connacht's first try came in the 18th minute as, on a Dragons put-in, the Connacht pack won the battle which brought Muldoon his first try that Parks converted.
It did not help that the Dragons were reduced to 14 men when lock Robert Sidoli was penalised at a ruck and spent 10 minutes in the sin-bin.
As such, the troubled pack were depleted again which spelt trouble at a Connacht lineout 10 metres out and saw Ah You driven over by his team-mates with Parks again adding the extras.
The Welshmen found some composure during half-time and came out with more focus but they struggled to get past the Connacht defence.
Muldoon's second touchdown, from a driving lineout on the right, put Connacht 16 points clear with 13 minutes left but the home side tried to get something out of the match.
Prydie roared over in the corner with 10 minutes left as the Dragons gained momentum.
But captain Muldoon kept his troops calm and Connacht had the last laugh when Swift went over for the bonus point score with seconds left.
- Sports & Recreation
- Jason Tovey
- Dan Parks