Marvin Emnes raced clear in the 89th minute to net Boro's winner just when it looked like the points would be shared after Matthew Lowton`s first-half header had earlier cancelled out Matthew Bates' 18th-minute strike.
The result leaves the Blades bottom of the Championship table, seven points from safety, whereas Middlesbrough are now 11 points clear of the relegation zone after they extended their unbeaten run to five matches.
Mowbray said: "It's great that we've got a victory. I think they feel hard done by.
"Sheffield United are still very much fighting and it won't be through lack of desire if they fail to pick up the points they need. We knew we'd have to come here and compete and ultimately we came here to do that job. I always felt we had quality on the break and we could score.
"We've played some nice football over the last few months. The players are fighting for each other and they had to do that against a determined Sheffield United team. It'll be quite a different game at Ipswich on Tuesday."
Source: PA
Source: PA