A Connacht player who has never played for his country before scored the match-winning try to help his club secure their first victory of the season.
The Ireland international found room on the left flank late in the game and crossed behind the posts to complete a dream start for Frawley.
Benetton had taken the lead through two tries by Leonardo Marin, but Connacht fought back after a tight period following a high-scoring opening quarter.
Sean O'Brien added to the Irish tally seventeen minutes into the contest with a well-worked try that followed smart play from Francois van Wyk and debutant Frawley.
Leonardo Marin and Frawley exchanged converted tries five minutes before half-time, leaving five points separating the sides at the interval.
Cian Prendergast opened the scoring after just five minutes by catching a quick pass from Sean Jansen out of a ruck to cross the line for his first try of the campaign on his first start.
Benetton replied almost instantly through Bautista Bernasconi after the front-row forward levelled from a maul following the early Connacht advantage.
Wayne Pivac's side started the second half brightly with Bernasconi scoring his second try of the game after more impressive forward play from the Italians to put the hosts in front once more.
Following the physical battle against the brilliant Benetton forwards, Connacht showed improvement from their scrum performance against Stormers in their opening loss.
The match took place at the home ground of the Italian club with replacements including Maile and Spagnolo for Benetton alongside de Buitlear and Bohan for Connacht.