The hard work was done in the opening half hour as Connacht were rewarded a penalty try 20 minutes in before George Naoupu went over, with Dan Parks adding ten points with the boot.
Despite the Dragons, who have now lost four straight RaboDirect PRO12 clashes, putting up more of a fight in the second half, man-of-the-match David McSharry added a try for Connacht as they eased to victory.
And ahead of the Irish province turning their attentions to Europe once more, Elwood was delighted to leave RaboDirect PRO12 duties on a high note.
"We put ourselves in a really strong position in the first half," he said. "We dominated possession, played some great rugby and got those two tries.
"With George's [Naoupu] injury we lost that little bit of momentum with the break in the game. There was a swing and Dragons got a push at the end of that first half. But we went in with a positive frame of mind at half time.
"I'm delighted that all the hard work the lads are doing paid off to get them the reward.
"Yes, we're a little bit disappointed not to get the bonus point. But at the end of the day, we got a good victory at home, it was a lovely evening for rugby and we had a great crowd of supporters."
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