The Teessiders dumped derby rivals Sunderland out of the Capital One Cup in the fourth round in October and gave eventual winners Swansea a real scare in their quarter-final tie at the Liberty Stadium before going down to the only goal.
Chelsea come to town in the FA Cup fifth round on Wednesday evening with Manchester United awaiting the winners in the last eight. And Mowbray said: "The Premier League teams we have played this year, we acquitted ourselves pretty well in both games."
He added: "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. 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's side will hope to throw off the shackles of a difficult run of results, which has seen them win just one of their last nine npower Championship games and slip out of the play-off places as a result.
They will have to do so without the services of the injured Jonathan Woodgate and Lukas Jutkiewicz - and with Josh McEachran, Kieron Dyer and loan signing Sammy Ameobi all ineligible.
The Boro manager freely admits he has no idea what side opposite number Rafael Benitez will field as he attempts to keep alive his hopes of winning a trophy this season, but knows whichever 11 men take the field, they will have quality in abundance.
Mowbray said: "When you manage a club that expects to win every competition it enters, or be very competitive, it's a big game for Chelsea. That's why I think they will come with a very, very strong side and try to get the job done on the night. The last thing they would want is a replay.
"I have to say, the last thing we would want is a replay as well because the league games are ultimately the most important thing. In the context of our season, our target is to get out of this league. The last month or so has been hugely disappointing for us and yet we can take this game in isolation and be positive."
- Sports & Recreation