The Championship club have already beaten Sunderland in the Capital One Cup this season, before narrowly being knocked out at the hands of eventual winners Swansea City.
Mowbray is confident that Boro can give the Blues a really competitive match.
"Here we are, we have beaten Sunderland and why can't Middlesbrough give Chelsea a really tough game and be competitive? I am sure we will be," Mowbray told ESPN. "The Premier League teams we have played this year, we acquitted ourselves pretty well in both games.
"We went out in the quarter-finals away at Swansea to the eventual winners in a game from which we felt really harshly done by not to come out with a positive result. Early on in that game in particular, we were on top and had the better chances. And Sunderland, to go to the Stadium of Light and win was a good day for the club.
"It shows the players they can do it, they can go and compete because if Chelsea were playing at Sunderland tonight, you would all expect Sunderland to give them a really tough game and be competitive."
Middlesbrough host Chelsea in the FA Cup at the Riverside Stadium on Wednesday.