Eleven points from the boot of Ben Botica and a try from Matt Hopper was enough to set up the clash against Bath. O'Shea said: "I'm delighted with the way we played in difficult conditions and delighted for the players that we have earned a place in the semi-finals."
He continued: "I thought we were outstanding in the first half and continued to play despite being unable to get any purchase from the pitch. They got a bit narrower as the game went on but that's understandable isn't it on a surface like that.
"All in all, it was a great learning experience for a young side and though Bath will come down to us with a full first team, we will play the same side again. These guys deserve it and they can look forward to that game.
"What pleases me is that we are the only unbeaten side in the last four and we are in the quarter-finals of the Heineken Cup.
"We have got a busy time coming up, but this is where we want to be. The guys want to win these competitions."
Ospreys coach Steve Tandy was taking the positives from a campaign that was over for the Welsh side before this game. Fly-half Matthew Morgan kicked all of their points and, while that helped them draw to within a point after 55 minutes, the hosts were unable to grab the lead.
"We did well but we couldn't get the foothold that might have led to a potential score to win the game. But they will learn from this. Harlequins are a top-class side," he said.
"Eventually you do need to win games like this, but I am immensely proud of the boys today. We had a lot of young boys out there, including Dafydd (Howells) on the wing who is only 17.
"When you consider also that we had five tight heads out too, I think we can take a lot from the game.We need to turn our attention back to the Pro 12 now and the trip to Ulster on Friday. It won't be easy but at least the likes of Duncan Jones, George Stowers and Jonathan Thomas have got some more rugby under their belts and that was important too."