Cameron Jerome's late deflected strike put the Blues in pole position to join neighbours Wolves in winning promotion from the Championship after they opened up a four-point gap over Kevin Blackwell's Blades.
United travel to face Burnley on Monday night and will realistically need a win to keep their hopes alive.
McLeish said: "It gives us a platform and keeps us in the driving seat.
"It puts the pressure on teams below us to get a result.
"I know how Sheffield United will be feeling. It is tough to respond to pressure and the more you handle the mental side of the game the better you will perform.
"But we have a tough task next week. We have to play Preston who had an amazing result by beating Cardiff 6-0 and they will come at us with full confidence.
"We are carrying a few knocks and one or two may be struggling but there are two games left and players can find fitness when they need to."
Jerome came off the bench to fire in the winner in the 73rd minute at Vicarage Road and McLeish admitted it was a tough decision to leave the former Cardiff striker out of his starting XI.
The Blues dominated the first half but were unable to beat Watford goalkeeper Scott Loach until Jerome lashed in a shot which looped in off the heel of Ross Jenkins.
McLeish said: "The goal was a very welcome sight indeed and we waited a long time for it.
"Jerome is an explosive player and he can't play every game as he loses his edge. We try to preserve that so that's why he didn't start but I told him there was every chance he will come on.
"He was fine about it. Every player is disappointed when they are left out but he took it in the right spirit.
"It's all about the bigger picture and we are all in this together."
Watford boss Brendan Rodgers felt his side were on top before Jerome's
goal but hailed Birmingham's resilience.
The decision to bring Grzegorz Rasiak on at the break transformed the Hornets after they had been under siege in the first 45 minutes.
Rodgers said: "In the first half it was difficult for us to retain possession up front and we never got going.
"But in the second half we were excellent and up until the goal there was only one tram who looked like scoring.
"I am disappointed we didn't take something from the game but congratulations to Birmingham.
"They are a good side and kept a clean sheet and took the three points.
"They know they have had a tough game but they are back on the bus with a win and that is the sign of a good side.
"When you lose to a deflected goal that wasn't going in it makes it even worse."
Rodgers admitted he faces a summer of battling to keep his best players as the Hornets attempt to cut costs at Vicarage Road.
He said: "There will not be a great deal of money this summer but that's the climate we are in. But that was the condition when I came in and I knew that.
"I want to keep all my players here but if an offer comes in which the club feel they need to keep the club afloat then we will have to look at it.
"But I will fight for them all."