Celtic had Stephane Mahe sent off by referee Hugh Dallas in the 43rd minute after tangling with Dundee's Eddie Annand and had to be led from the field by his colleagues as he protested his innocence.
But Celtic, struggling after five defeats in their previous eight games, kept Dundee at bay and, with extra time looming, the substitute Mark Burchill sped down the left and crossed for Wieghorst to place a looping header into the net.
Wieghorst gained a recall at the expense of Eyal Berkovic, who was a surprise omission from the Celtic starting line-up, despite no previous indication that he was injured.
Barnes shrugged aside the departed Craig Burley's view that he could do better at Derby County than at Celtic and praised his team's efforts without the Scotland midfielder. "The 10 men who won it for us tonight deserve all the headlines and plaudits," he said.
"It was an excellent performance and the players deserve a lot of credit. Their response to the sending-off was superb and we merited the win at the end."Reuse content