A pulsating match which saw a stack of chances and excellent goal-keeping at either end looked to be heading towards a goal-less draw until the 79th minute, when Dons midfielder Jeffrey de Visscher linked with Lee Miller at the edge of the box before driving low past Hoops goalkeeper Artur Boruc.
The Dons were looking forward to the semi-final meeting with Irn-Bru First Division side Queen of the South next month, but in the second minute of injury time Vennegoor of Hesselink threw himself at a cross from substitute Georgios Samaras to force the ball over the line from a couple of yards out.
Strachan, who revealed that defender Lee Naylor’s last-minute replacement by Darren O’Dea was due to illness, said: “I thought the draw was a fair result. We hit the post three times, their goalie was excellent and our goalie was excellent.
“So I think both teams deserve another shot at it for the hard work they put in.
“We try and give our fans entertaining football and that was an entertaining game. In terms of cup ties this season that might be the best that you’ll see.”
Aberdeen boss Jimmy Calderwood was left feeling like his side had been knocked out of the cup but insisted they still have a chance in the replay at Celtic Park on Wednesday week.
“It feels like a defeat at this moment,” he said. “We started well, we kept Aiden McGeady and Shunsuke Nakamura quiet and I think we deserved to be ahead at half-time.
“At 1-0 ahead we were confident and Artur Boruc made two or three great saves in the last couple of minutes.
“If Scott Severin gets the ball away at the end then it is all done and dusted but he mis-kicked it and it ended up at the feet of Samaras.
“He had the quality with his cross and Vennegoor of Hesselink has done us with 30 seconds to go.”
Soutar, Maybury, Considine, Diamond, Foster, De Visscher, Severin, Touzani, Jamie Smith (Aluko 31), Miller, Maguire.
Subs Not Used: Langfield, Mair, Bus, Walker.
Boruc, Hinkel, Caldwell, McManus, O’Dea, Nakamura, Hartley, Sno (Donati 77), McGeady, McDonald (Samaras 72), Vennegoor of Hesselink.
Subs Not Used: Mark Brown, Pressley, Riordan.