The goal and Cox's mistake turned the game on its head as until then Watford had been in the driving seat and had taken a deserved 2-0 half-time lead.
The Hornets had squandered two fine chances early on against a lacklustre Leicester side.
Heidar Helguson had been set up by a fine cross from Lloyd Doyley only for the Icelandic striker to head wide when unchallenged.
Watford's Danny Webber also missed a good chance when he cut in after a pass from Johnnie Jackson and beat Ian Walker with a powerful shot but the ball struck the far post and bounced out.
So it was no more than they deserved when Watford took a 40th minute lead. Neal Ardley sent over a free-kick and Jay Demerit was there to head firmly past Walker. Three minutes later City's Patrick McCarthy made an unnecessary challenge on Helguson in the box and Webber converted the subsequent penalty.
After Connolly had put Leicester back in the hunt early in the second half City dominated the proceedings. Watford were unable to make any progress although Webber did hook the ball just wide on the turn with Walker seemingly beaten.
But Connolly, the object of derision from the home fans, was there to set up the equaliser after 71 minutes. He cut the ball back for Stephen Hughes to fire home.
A few minutes later Connolly looked set to score the winner but Lee made a fine blocking save.