Johnson was in charge of west country rivals Yeovil when the then-Conference side thrashed league opposition in the shape of Colchester 5-1 at Huish Park in the first round in 2000 before beating Blackpool 1-0 away in round two.
Three years later and Johnson repeated the trick with comprehensive 4-1 and 5-1 home victories against Wrexham and Barnet respectively at the same stages of the competition.
But the former Latvia national coach is adamant victory over the Premiership side on their own turf would top them all.
He said: "Obviously the things we did at Yeovil provide great memories and are terrific achievements in their own right.
"But I think to beat a Premiership team in their own back yard would compare more favourably, especially when you consider the gap between the top flight and the other divisions these days.
"At Yeovil we were beating league sides that would now be in the equivalent of League Two and League One and the gap in terms of money and the quality of football was not that big.
"But now, for League One Bristol City to go to Middlesbrough and win in the FA Cup, that would be a huge achievement."
The Robins head to the Riverside looking to claim a Premiership scalp in the competition for the first time since Russell Osman's side beat Liverpool 1-0 in a third-round replay at Anfield in January 1994.
Johnson added: "Obviously we will be the underdogs but we don't mind that.
"It will be a tough game for us but hopefully we can go there an make it tough and uncomfortable for them too."
Gareth Southgate said
"Bristol City showed great spirit at Ashton Gate and it was not the first time they had hit back from a couple of goals down this season.
"They had not lost at home in ever so many games and so we had a pretty good idea what to expect.
"We went there intending to play our game but Bristol City put us under a lot of pressure in the second half."
Bristol City have failed to win any of their three league games since that cup tie, with Lee Johnson's late leveller against Huddersfield on Saturday the only goal they have scored in those games.
Southgate added: "Obviously Bristol City haven't had the best of results since we played them but they are involved in other cup matches and it is a hectic time for them.
"Any side of Gary Johnson's is going to be difficult to play against and we will give them the greatest of respect again tomorrow.
"This is a no lose game for them and so we have to respect them and prepare for another tough game.
"But we have home advantage and it's a game we should expect to win. We have to do everything it takes to make sure we do win.