Sunderland, famous giant-killers themselves when they beat Leeds in the 1973 final, were held by Sky Bet Championship side Fulham in a goalless draw at the Stadium of Light at the weekend and the two sides will meet again next week.
Lawn, co-chairman Julian Rhodes and Bradford manager Phil Parkinson had all wanted to be paired with Manchester United after Saturday's stunning triumph over Chelsea at Stamford Bridge.
But Lawn had no complaints after Bradford were drawn to play either Sunderland or Fulham at home in Monday night's draw and has good reason to cheer on Gus Poyet's Black Cats in their replay.
"If we can't get Man United, Liverpool or Arsenal, let's have a home draw
It's up to us now," Lawn told Press Association Sport.
"We're the lowest-ranked team left in it, apart from Cambridge, and the odds are stacked against us, but let's give it our best shot.
"If we play like we did against Chelsea we will take some stopping.
"I would prefer Sunderland because of my mother
She was a Mackem
I lost her in 1993 and she was one of 13 children, so imagine how many cousins I've got!
"She moved down here just after the war and worked in Harry Ramsden's
That's where she met my dad."
Sky Bet League One side Bradford's 4-2 win at Chelsea is one of the biggest cup upsets of all time and follows just less than two years after Parkinson steered the West Yorkshire club on a fairytale journey to Wembley in the Capital One Cup final.
Parkinson led the Bantams to promotion via the League Two play-off final later that season and although he was rewarded with a new three-year contract soon after, the club's latest cup exploits will inevitably lead to questions over his longer-term future.
"He's still got 18 months left on his contract so everyone can bugger off!" Lawn responded.
"It would have to be something very special to take him from here
His family live in the area and I'm sure they're all very happy here.
"That would end up being down to Phil, but of course we want to keep him."
Source : PA