Treviso, whose squad has been disrupted with several World Cup call-ups, started brightly and twice came close to scoring the opening try and took a deserved lead courtesy of a Willem de Waal penalty after eight minutes.
The South African fly-half then had the opportunity to double the hosts' advantage but missed what seemed an easy penalty.
The miss from de Waal seemed to galvanise Connacht who were level after 28 minutes when Matthew Jarvis scored a penalty of his own.
The visitors were in front at half-time thanks to Ah You's try with the New Zealander going over after some good work by lock Michael Stewart, although Jarvis missed the conversion to make the score 8-3 at the interval.
An attritional second half lacked clear-cut opportunities but Jarvis' second penalty after 56 minutes put Connacht further ahead and they held on to win the their first game of the new season.
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