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A stunning first-half score from Tom Guest and a 51st-minute effort from Mike Brown proved the difference between the two teams in the top-of-the-table clash in Pool 1.
The sides meet again at The Sportsground in just five days time and O'Shea is expecting an even tougher test at one of the continent's most revered venues.
"I'm really looking forward to it. It's a really tough place to go," said former Ireland full back O'Shea.
"I've gone down there through the years as a player and a coach and it's a heck of a tough place to go.
"But the really good thing from our point of view is that the lads are forewarned. We played them in pre-season and it wasn't the way Connacht play. Tonight was the way Connacht play.
"It's going to be really tough, as I said. They've got good ball players all over the place and they'll take a lot of confidence from tonight. We'll take a lot of confidence, too, in that they had a lot of possession and we didn't concede a try."
Quins now sit five points clear of their Irish rivals at the halfway mark of the pool stages after two wins and one defeat in their three games to date.
Denying Connacht a losing bonus point could yet prove crucial in terms of qualification but O'Shea insists winning was the only thing on his mind tonight and that will remain the case on Friday.
"Bonus points here are irrelevant. These are two head-to-head games," added O'Shea.
"We've won the first one and we have to go out and win the second one - then everything's in our own hands.
"We know exactly what we need to do in Connacht. We'll just and freshen everyone up over the next few days and we'll look forward to the trip to Galway."
