Gavin Skelton's powerful strike rocketed off the bar and bounced in off Pascali in the 82nd minute to earn Killie a 1-0 win and a fifth-round home tie against Inverness.
Calderwood, whose first match ended in a 3-0 home defeat by Motherwell, missed the ball going in because he still had his head in his hands after Skelton had struck the bar.
But he hopes the instant change in fortune will turn out to be more lasting.
"As in most times, the first goal was always going to be vital and we got it," Calderwood said. "I didn't think we were going to get it. It looked like we were going to have another game and that's the last thing we wanted.
"It was a good corner and came off the underside of the bar but Pascali got it in off his knee.
"I didn't know a lot about it. I thought it was bad luck and thought 'Jesus, we've hit the crossbar'.
"Then I saw them jumping about. I couldn't see but it seemed like we had a bit of luck.
"Time will tell, but maybe that is the wee break we needed."
Calderwood also praised Cammy Bell for making several excellent saves, including a one-handed stop from Vitor Lima in stoppage time.
But Falkirk manager Eddie May admitted his side only had themselves to blame after missing several good first-half chances, especially when the unmarked Jack Compton blazed over from 14 yards.
May said: "Credit to Kilmarnock - the chance came along and they took it. But we had opportunities, certainly in the first half, to put ourselves in a comfortable position, and we never took them.
"But the boys gave everything they had. I think they played half-decent in spells in the first half and caused them some problems, and we should really have scored. Their goalkeeper had two good saves."
May added: "Jack should have scored, with the quality he has. I'm sure he didn't mean to put it in row Z, he just has to be a little bit more composed next time. He probably hurried a bit too much."
Source: Team Talk