Conor Newton had the Scottish Premiership side ahead after just nine minutes at Palmerston but McGowan limped off soon after with a hip injury.
The former Celtic man's departure hindered the Paisley men and the Championship Doonhamers levelled the fourth round meeting 10 minutes before the break through Iain Russell.
Steven Thompson did grab his seventh goal of the season to put Lennon's team in front once more just after half-time but they lacked the potency in midfield to keep hold of possession.
And that allowed Queens to secure a rematch at St Mirren Park on December 10 with another equaliser by Michael Paton 18 minutes from time.
Lennon said: "Gowser picked up a knock in training but decided he couldn't go on after a quarter of an hour and asked to be subbed. We actually missed the wee man when he came off but I have to respect his honesty as he put the team first.
"He would not have been his usual busy self if he had stayed on but it was an important point in the match."
League Cup holders Saints saw their defence of that trophy ended at the first hurdle when Paton grabbed an extra-time winner the last time the sides met in August.
But Lennon felt his side had learned their lesson.
He said: "We came down for the cup game and were bitterly disappointed to go out so early as holders.
"But coming down here again that result was there as a warning and we more than matched their energy this time.
"I'm a wee bit disappointed that we got ourselves in front twice but didn't hold on to it for a prolonged period of time.
"That is all down to an energetic Queen of the South side.
"It was a proper cup tie. It wasn't great football played by either side but both sets of fans were thoroughly entertained."
- Sports & Recreation
- Michael Paton