Boro head for Burnley on Tuesday evening having lost six of their last seven Championship fixtures and each of their last six games in all competitions on the road, culminating in Saturday's 4-1 reverse at Crystal Palace.
Mowbray told the club's official website: "It's incredible after the run we have been on that we still find ourselves in the top six and in the fifth round of the FA Cup - but we have to start winning football matches.
"I have said to the players I'm pretty confident we will. If we don't make the top two, hopefully we will make the top six.
"I'm pretty sure we will finish with a strong run, hopefully starting at Burnley. The mood is still positive.
"I tried to give them (the players) an analogy about a boxer getting back up and throwing the first punch again, and I'm sure we will do that and that at the end of the season, we will be looking back on a sticky period in January and February."