Monday evening's draw pitched Chelsea's unlikely conquerors into a home tie with either Gus Poyet's Premier League Black Cats or Kit Symons' Championship side, who are due to replay next week after a 0-0 draw on Saturday.
While most Bradford fans were hoping for a tie with Arsenal or Manchester United, a home clash will be welcomed and Lawn is of the opinion that neither Sunderland nor Fulham will fancy the trip to West Yorkshire on the weekend of February 14/15.
"We're at home
I think we can win at home," he told Sky Sports News
"They won't want to come down here and get this lot (the fans) bouncing
I don't think Sunderland or Fulham will want to come to Valley Parade."
City's 4-2 win at Chelsea - having been 2-0 down to the Premier League leaders - is already one of the cup's greatest upsets and another feather in the cap of manager Phil Parkinson.
The former Charlton boss oversaw Bradford's run to the League Cup final in 2013 and earlier this season took the scalp of Leeds in the same competition.
Lawn, though, is confident that his man will stay at the helm despite the interest he is likely to generate.
"He's got 18 months left to run and we don't want him to go anywhere," he added
"I think he's a bit rubbish, so all the other teams, keep your bloody hands off!"
City last faced Sunderland back in December 2003, when they lost 3-0 with a side containing the likes of Wayne Jacobs and Dean Windass
Their last meeting with Fulham came further back in 1994, a 0-0 draw.
Bradford were not the only side to cause a shock at the weekend, with Middlesbrough's 2-0 win at Manchester City equally worth of praise
Aitor Karanka's Championsip high-flyers will now head to holders Arsenal, who beat Brighton on Sunday.
League Two Cambridge, who held Manchester United to a home draw, go to Old Trafford next week with the winners headed for a tie against Sheffield United or Preston
North End boss Simon Grayson has met Manchester United before, knocking them out while with League One Leeds.
In-form Crystal Palace await the winners of the replay between Bolton and Liverpool, while Leicester and Aston Villa square off in a midlands derby
West Brom - victors over Birmingham - are at home to West Ham, who rode a storm against Bristol City on Sunday.
Blackburn lie in wait for the winners of Rochdale v Stoke, while an all-Championship tie between Derby and Reading wraps up the draw.
Source : PA