The League One Bantams were on their way out after falling two goals behind after Gary Cahill and Ramires netted for Chelsea.
However, Jon Stead gave the visitors hope by halving the arrears shortly before the break
Felipe Morais equalised and Andy Halliday gave them a sensational lead.
Premier League leaders Chelsea pushed for a late leveller, but Bradford broke and ensured their place in Monday's fifth-round draw through Mark Yeates' 90th-minute strike.
"This is an ultimate, to beat Chelsea, who I think will go on and win the Champions League - it really is," Lawn told BBC Five Live Breakfast.
"I've been saying it's like beating Barcelona
Someone texted me and they said the side they put out against us was worth Â£98million and ours was worth Â£7,500.
"I am a bit stunned still, I'm not pinching myself in case I was dreaming.
"Man United, Liverpool (in the next round) that would be nice."
Six thousand Bradford fans made the trip to the capital from West Yorkshire and Lawn paid tribute.
"They were the difference, they kept us in it," he said
"We were 2-0 down but they kept pushing the players, can you tell me another team that has put four past Mourinho in his career?"
And two Bradford fans, Tom Fletcher and Dominic Newton-Collinge who run the Bradford Banter podcast and blog could not contain their delight.
"We've been lucky enough to see cup runs, Wembley wins, but yesterday was unbelievable the best result I have ever seen of any football game I have ever watched," they said.
"We took 6,000 fans and not one sat down in the entire 90 minutes and every one sang their hearts out
We made that feel like a fortress for Bradford City even when we were two goals down.
"All four of our goals were of the highest standard, it was unbelievable to witness, every one of those goals was amazing
"It is one of the hardest things to describe how that feels as a fan of a League One team
"To go to the best club in the country and not only to come back from two goals then to get a third, to get into seven minutes into added time and think 'Chelsea are going to score, that's what happens in football' and then utilise that seven minutes to your own advantage and score another, we almost got cocky!"
Source : PA