The Quins took some time to subdue an enterprising Irish side and were fortunate to lead 6-3 after the first half an hour. Their set scrum then emphasised their superiority to ensure there was no way back for Connacht, who were briefly reduced to 12 men in the second half as they fell to a heavy defeat.
"I am really delighted, we are almost guaranteed a home quarter-final," said O'Shea.
"Connacht came to play rugby; it was tough for long periods especially as we played into a stiff breeze in the first half. The penalty try just before half-time gave us breathing space.
"We were far more direct in the second half and played some superb rugby."
Connacht counterpart Eric Elwood reflected on some missed chances for his side, saying: "We had some great opportunities in the first half-hour and didn't have any reward.
"We had four or five great line-breaks and didn't finish them off.
"How we were 6-3 down after 35 minutes I just don't know, but Quins scored just before and after half-time to build up a cushy lead."
Elwood also bemoaned the loss of three players in the second half to the sin-bin.
He added: "The referee was particularly harsh, Johnny O'Connor was a nothing affair and (Fetu'u) Vainikolo's was a 50-50 tussle for the ball with Ugo Monye. It's impossible to defend with 12 or 13 players."
- Sports & Recreation