The Quins fly-half was spotless from the tee all night, converting both of his side’s tries as well as potting six penalties to inflict a first defeat on Exiles in the competition.
Quins stormed into a 29-3 half-time lead at Zayed Sports City, with Jacques Benade’s men paying for every mistake.
Stevenson knocked over two early penalties and even though Durandt Gerber reduced the arrears with one of his own, that was as good as it got for the visitors in the opening period.
The hosts scored the first try when winger Brook Keats went over after great work from centre Brian Geraghty, Stevenson extending the lead to 13-3.
He added another penalty as Quins got into rhythm, flanker Ben Bolger dotting down for their second try after a powerful forwards surge from a lineout.
Stevenson converted from a difficult angle and the home side were rampant.
Exiles continued to transgress and Stevenson kicked two more as the score stretched to 29-3 and as the half-time whistle blew, Exiles were facing an uphill struggle.