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.
Now Mowbray is challenging his players to prove their credentials once again when Chelsea come to town in the FA Cup fifth round this evening with Manchester United awaiting the winners in the last eight.
He said: "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.
"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."