The Championship outfit had put themselves in poll position to pull off a major upset by recording a 2-0 half-time lead over City, with Fernando Forestieri and Deeney on target for the London club.
However, Sergio Aguero found a way through the Hornets' stubborn defence on the hour mark, before leveling matters on 79 minutes.
With three minutes remaining, Giuseppe Sannino's men looked set to hold on for a replay at Vicarage Road, only for Aleksandar Kolarov to fire the Blues into a 3-2 lead before the Argentine completed his hat-trick in the closing stages to end Watford's resistance.
"We played well. It was so close but they showed what it takes to be in the Premier League, the ruthlessness they showed and that's what we need," striker Deeney told reporters.
"People said it was going to be 10-0 and nobody gave us a chance but we knew we are not a bad set of players.
"City are one of the biggest teams in the world but we showed no fear. They had four forwards on, who knows how much they all cost, but we gave it a real go."
The Sky Blues discover who they face in the fifth round of the competition tomorrow evening.