Thistle needed Scott Fox's penalty save from Jamie Stevenson to survive 90 minutes before extra-time goals from substitutes Christie Elliott, Steven Lawless - either side of a Kane Hammeings equaliser - put them in front.
Aaron Muirhead sealed a 3-1 win with four minutes left and Archibald was relatively unconcerned by the lateness of the win.
"I'm just delighted to get through," he said. "All credit to Cowdenbeath, I thought they were fantastic, they had a game plan and stuck to it and sat behind the ball.
"We knew they would always get a chance and thankfully Scott Fox had an outstanding save from the penalty."
Cowdenbeath manager Colin Cameron felt his part-time players naturally ran out of steam towards the end of extra-time.
"We worked very hard and got the wee break from the penalty and Jamie is normally lethal from there," he said.
"We go into extra-time and to our credit get a goal behind and get right back on it.
"Foxy made an unbelievable save from Kane Hemmings and John Armstrong had a header that went just by.
"Then maybe just a wee bit of tiredness told, and the penalty at the end seals it, but the only negative was losing."