Despite dominating the first half, the Blues looked to be heading for a fifth draw in seven games until manager McLeish introduced Cameron Jerome for Gary O'Connor with 20 minutes remaining.
Within three minutes, the change had worked.
The former Cardiff City striker latched on to a ball on the left side of the penalty area and after stepping inside Watford skipper Jay DeMerit, hit a right-footed shot that took a wicked deflection off Ross Jenkins and gave keeper Scott Loach no chance.
It was tough on Brendan Rodgers' men, and Loach in particular, after they had recovered from a being a distinct second best in the first half to take the game to their opponents after the break.
The Blues had taken the game to the home side from the opening whistle and they proceeded to dominate the opening period, only to find Loach in superb form.
James McFadden, O'Connor, Sebastian Larsson and Keith Fahey, twice, had already warmed the England Under-21 international's gloves from varying distances, but five minutes before the break the keeper made a good save with his legs to deny Larsson from the right side of the penalty area.
This was soon followed by another fine stop to keep out a Lee Bowyer shot from a similar position.
Watford 's first-half attacking efforts were restricted to a long distance effort from Lloyd Doyley and Jobi McAnuff's near-post flick, but Rodgers made two changes at the break.
The fit-again Grzegorz Rasiak was introduced to add a more physical dimension to the Hornets' attack, while Tommy Smith was moved in field to support the Pole.
Twice within the opening five minutes of the second half the adjustments nearly paid off, with Danny Rose narrowly failing to turn a cross from Rasiak on target before the on-loan Southampton striker was just unable to get off a cut-back from Smith.
That set the tone for a more even second period with Watford again going close to taking the lead when Colin Doyle clawed away a Rasiak header that was destined for the bottom corner following a McAnuff cross.
With the Blues' attacking threat diminishing though, McLeish brought on Kevin Phillips and then Jerome.
The second change, coupled with a slice of good fortune, was to prove pivotal.