An out-of-sorts Quins team trailed 15-3 after an hour and looked certain to suffer their first defeat of the season, but two converted tries - the first from Sam Smith, and the second a penalty try - turned things in the visitors' favour.
"Worcester must be gutted to lose," O'Shea said. "We were very unhappy with our performance today.
"The game was a very start-stop affair it was free-kick after free-kick and penalty after penalty and for 60 minutes we were awful.
"The guys let themselves down but at 15-3 down we could have waved the white flag. It is very satisfying that the boys dug in and secured the win."
For the first three quarters of the match Harlequins looked nothing like league leaders as they were shunted out of their stride by some ferocious tackling by the Warriors.
"Worcester made it very difficult for us they were incredibly physical, up in our faces and hitting every tackle with venom," O'Shea added.
It was a disappointing match with neither side playing remotely near their best and, like O'Shea, Worcester head coach Richard Hill was also critical of his team.
He said: "At 15-3 up at home you have got to win these games we were in control but we gave away a soft try from a first phase and credit to them for coming back to win.
"It was a poor game. In the first half we made eight errors in the first 17 minutes and our lineout was a huge concern.
"We couldn't keep the ball for sufficient phases once we had five phases then four and we got two penalties so it is not rocket science."
Worcester suffered a blow when influential captain Kai Horstmann left the field with his side 15-3 up.
Hill reported afterwards: "Kai has pulled a hamstring. It doesn't look good and he will be out for four to six weeks."
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