Despite leading 9-6 at half time, Quins were eventually edged out 18-12 at Twickenham Stoop.
"The better side won," O'Shea told London24. "They won the breakdowns and the collisions and we are bitterly disappointed.
"We did not play the way we could today. We were not allowed to play because Munster bossed the second half."
Quins continue the defence of their Aviva Premiership title on Saturday when they entertain Bath.