And mid-way through the second, man-of-the-match Muldoon boosted the advantage with his second try of the day.
The only consolation was a 69th minute try from winger Tom Prydie who took advantage of a short-benched Connacht following Darragh Leader's yellow card a minute before.
The bonus point win for the Irish side leapfrogs them into seventh in the RaboDirect PRO12 while Newport fall to ninth, only ahead of the Cardiff Blues and Italian sides Benetton Treviso and Zebre.
Jones insists the Dragons - playing their first league match since last month - need to cut down on errors if they are to bounce back against Zebre this weekend.
"We have got a lot of work to do but the two weeks off we have just had cost us dearly," he told WalesOnline. "We lost our shape and momentum and we need to learn from that.
"It's not difficult to see where our weaknesses are but this is what we've got here and we have to deal with it. Connacht played well, their error count was low but we were pretty poor.
"Sadly that was the worst opening 20 minutes that we have had all season. It took the wind out of our sails and killed off any momentum that we had tried to create.
"It's a shame because we've made so much progress throughout the season. We've taken a big backward step here today in many facets of the game."
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Guinness PRO12 Dragons News
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