Boro gave as good as they got for long periods and while their FA Cup dream may be over, Mowbray was hoping to take some of the positives into the remainder of their campaign, which continues against leaders Cardiff at the weekend.
It took Chelsea 51 minutes to force their way in front when Yossi Benayoun recycled Oscar's cross to Ramires and he curled the ball towards the top corner.
His shot flicked Fernando Torres on its way to goal and both men appeared to claim the goal, although few in blue would have been unduly concerned as to the name which eventually went on to the scoresheet.
The visitors made sure of victory 15 minutes from time when substitute Eden Hazard laid a second on a plate for Victor Moses, who capped an impressive individual display by tapping home.
Mowbray said: "We have got 12 games left now. For six months of this season, we have been in the top six and we have been threatening the top two for most of that time.
"We have obviously fallen off the pace in the last six weeks, for whatever reason, be that injury or the fact we have been involved in cup competitions.
"But we have to get back to business now. We have to start winning football matches again now and our only concern this year is to try to get out of this league."