The Hoops lost 1-0 away to Ajax on Wednesday night to slump to the bottom of their Champions League section and leave qualification hopes up in the air.
However, van Dijk led the charge against the Staggies, heading in an Emilio Izaguirre cross past County goalkeeper Mark Brown in the 41st minute after threatening on three earlier occasions.
It was the 22-year-old's first goal in green and white signing from Dutch side Groningen in the summer for around £2million - and in the 52nd minute he got his second when he headed in a Charlie Mulgrew corner.
County substitute Ivan Sproule reduced the deficit in the 67th minute, after replacing Mark Klok, but midfielder Joe Ledley helped himself to a brace of headers to underline what was an impressive response from the Parkhead side to their midweek disappointment.
First-half goals from Nick Ross and Billy McKay secured three points from the Scottish Premiership clash, as Butcher looked on from the main stand.
The Englishman is tipped to be confirmed as Pat Fenlon's successor on Monday and opted not to take his place in the opposition dug-out for the meeting with his potential new club.
And he will have his work cut out for him when he completes the move after watching an Inverness victory that should have been even more convincing.
Hearts came from behind to seal a vital and impressive 3-1 victory over high-flying Aberdeen, but the Dons had only themselves to blame for throwing away a match in which Niall McGinn had given them the lead.
McGinn subsequently missed a penalty before midfielder Barry Robson was sent off in the 55th minute. Robson had only been on the pitch because injury had forced Willo Flood off after 22 minutes.
Jamie Walker hauled the visitors level in the 66th minute before turning provider for Calum Paterson to stroke home. Ryan Stevenson sealed the points in stoppage time.
The double by Thompson and one from Conor Newton were enough for Danny Lennon's men to take the three points back to Paisley.
However, the scoreline did not reflect the pattern of large periods of the match as Partick were again left with nothing to show for their dominance of a game.
Their misery was compounded with the dismissal of Stephen O'Donnell midway through the second half.
Two goals from Ryan Gauld and one each for Paul Paton and Andy Robertson saw the visitors to victory but they were helped considerably by poor defending from a makeshift home rearguard.
Nigel Hasselbaink opened the scoring before Stevie May netted a deadly brace, one from the spot.
Sean Clohessy's freak cross did reduce the final deficit to 3-1 for the Ayrshire outfit who had Jackson Irvine and Darren Barr red-carded in the second half.