Quins were cruising at the break as scores from Andre Esterhuizen, Marcus Smith and Cadan Murley replied to Scott Cummings' early effort.
Glasgow dominated the first half hour of the second half with tries from Johnny Matthews and ever-present George Horne levelling the contest, before Horne's penalty took Warriors ahead.
But replacement hooker Riley took his chance from the back of a maul on 76 minutes to send Quins into a Champions Cup quar ter-final for the first time in 11 years.
"Whenever we're ahead, we seem to let the opposition in," said star man Smith. "That's something we're working on. We're still not where we want to be, but we've got another European knockout now, which is exciting.
"In the last two years we've fallen at this hurdle, so that's a monkey off the back."
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