The Holland international had to be replaced by top scorer Georgios Samaras after clashing with Accies defender Brian Easton with the game only seconds old.
On the face of it, the loss of Vennegoor of Hesselink made little difference to the outcome as the home side responded to their Champions League defeat to Villarreal in midweek by cruising to the top of the Clydesdale Bank Premier League table on goal difference over rivals Rangers.
Goals from Shunsuke Nakamura and Samaras gave the home side a half-time lead.
Scott McDonald grabbed number three in the 75th minute and Aiden McGeady slid in the fourth to complete the visitors' misery.
But Gordon Strachan could not hide his disappointment at the injury to his Dutch striker who could be out for a number of weeks.
He said: "It is an abductor muscle injury, so that's not good news.
"That's the only downside to the day to be honest. I don't know how long he will be out. In terms of Jan's injury, it doesn't help me that I can bring our top scorer off the bench.
"He had been carrying a wee injury but it was something completely different. He doesn't know yet what happened. I thought it was his knee at first so I'm happy that it wasn't that."
Strachan declared himself "more than satisfied" with his side's performance against the SPL new boys.
"I am absolutely delighted. The attitude pleased me. It could have been a problem," he added. "It was windy, wet, we had just played a European tie and had been on the plane for a long time.
"To start the way we did with energy and taking it from there, that pleased me. All the players were terrific."