Everton scored twice in the last couple of minutes to snatch a well-deserved 2-1 victory from a Tottenham side who looked to have escaped the Goodison Park fortress with all three points.
However, Steven Pienaar headed an 90th-minute equaliser and Nikica Jelavic turned home the winner in added time to extend their unbeaten home league record to 12 matches.
Had they not got at least a point, it is certain Moyes would have been complaining about a penalty claim which did not go their way late in the first half when Pienaar's shot hit the arm of William Gallas.
Fortunately he did not have reason to complain after his team's late turnaround ended a dismal sequence of seven draws in nine Barclays Premier League matches.
With the injured Gareth Bale out of the team, Tottenham did not have the same attacking thrust down their left. As a result Everton were able to take advantage, particularly in the first half, by pouring forward down their right side.
Everton began to dominate without threatening. Shouts for a penalty against Dempsey after the ball came off his chest were magnified greatly when Pienaar's shot hit Gallas' arm but referee Kevin Friend was not interested.
Having soaked up so much pressure it was almost inevitable the visitors would produce a sucker-punch and it duly arrived in the 76th minute. Dempsey received the ball with his back to goal 30 yards out, turned and hit a shot which took a slight-enough deflection of Sylvain Distin to send it dipping over goalkeeper Tim Howard.
Everton's equaliser duly came in the last minute when Seamus Coleman crossed and Pienaar headed past Hugo Lloris, but there was further late drama as Darron Gibson's pass was touched on by substitute Apostolos Vellios and Jelavic struck to the delight of Moyes, whose exuberant celebrations saw him encroach onto the pitch.