But a hat-trick looked unlikely with ten minutes remaining as Gonzalo Garcia's long-range penalty had the Italians ahead by two points.
But home fly-half Dan Parks responded with a drop-goal and a penalty of his own to lift Connacht to a 23-19 victory.
Next up for Connacht is a daunting trip to Munster but after stringing together a fine run of form, Elwood believes his side will arrive at Musgrave Park with their heads held high.
"We were a bit nervy and people expected us to win, but thankfully we came out on the right side again," he said.
"It's been a good block of games for us and after three good victories, we relish the challenge of Munster next time."
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