The Chelsea manager had watched his side recover from a one-goal deficit dealt them by James Beattie's 36th-minute penalty and, after a second-half equaliser from Frank Lampard, saw a goal by his Ivory Coast striker just after the hour ruled out by referee Mark Clattenberg for offside. It provoked outrage from Mourinho, who even suggested the official should take the example of referee Rob Styles, who apologised to Chelsea for not awarding a penalty against Aston Villa last season.
The move in question had been orchestrated by Lampard, who nudged a ball through the Everton line from outside the penalty area. Waiting was Drogba, who struck his shot out of the reach of Nigel Martyn. While television replays revealed the striker to be on-side his team-mate Eidur Gudjohnsen clearly was not, although whether his presence was judged to be interfering with the passage of play was left to Clattenberg's interpretation.
"I am going home thinking we won three points," Mourinho said. "We scored two great goals, one was a fantastic shot, the other was a magnificent move through the middle. The second was a clear goal - if the linesman has doubts he has to give the advantage to the attacking team. There were no doubts for me - at the time and on the pitch it was a big mistake."
He may hope for an apology, but Mourinho will not be able to rescue a start to the season of nine wins and a draw, which means that the record set by Bill Nicholson's Tottenham in 1961 of 11 straight victories halted there after 34 matches, Liverpool's 1987-88 run of 29 games went the same way and three years ago Wayne Rooney's goal ended an Arsenal run of 30 games.
It will be little consolation for Mourinho who, unusually for the Chelsea manager, saw his team more than a little bullied by the side at the bottom of the Premiership for much of the first half.
Duncan Ferguson and the impressive James Beattie proved difficult for Mourinho's defence to marshal and, although John Terry and Robert Huth matched those two strikers challenge by challenge, there was a fresh uncertainty to Chelsea's distribution.
They were not permitted to build their attacks in the normal methodical fashion, there was pressure applied to Claude Makelele and the usual tempo of Chelsea's advance was disrupted. Shaun Wright-Phillips and Asier del Horno were presented with meaningful chances by Frank Lampard but failed to take them and the penalty that Chelsea conceded before the interval saw them in an uncharacteristic state of confusion.
It was Wright-Phillips, caught in possession by Tim Cahill, who chased back without discipline and clipped the Australian midfielder's legs just as he crossed into the penalty area.
Until then, Everton had not threatened Petr Cech's goal other than a David Weir header that the goalkeeper punched clear. With the memory of his disastrous sending-off after just eight minutes into this fixture last season, Beattie placed a confident penalty-kick high into the net
For at least 30 minutes yesterday the Everton striker will have contemplated what his penalty would have meant for the Premiership had Everton been able to defend their lead and they reached the interval with it still intact.
Half-time, however, has of late been the point at which Mourinho has transformed his side and, although this time there were no substitutions, it was a very different Chelsea side that emerged from the tunnel to take control of the match.
Perhaps Cahill's injury, picked up in a clash with his own team-mate Tony Hibbert, was the key to the shift in power in the midfield but there could be no denying the enormous influence that Lampard wielded in the second half.
This match had quickly become his show and in the fifth minute of the second half he allowed a throw-in from the left to run across his body before striking his ninth club goal of the season inside Nigel Martyn's left post.
It became a simple matter of keeping Chelsea out in the closing stages as Mourinho switched to a 4-4-2 formation with Hernan Crespo and the influential Gudjohnsen up front and sent on Arjen Robben to test out the right side of Everton's defence.
When Mourinho had finished protesting against Drogba's disallowed goal in the 63rd minute, he paid tribute to his side, who had "dominated the second half and had not stopped running until the last second."
Moyes could not celebrate any change in his side's status as the Premiership's bottom team but in response to Mourinho's protests over Drogba's disallowed goal did offer one of his own.
Substitute Marcus Bent's late shot appeared to strike John Terry on the arm, although all appeals were waved away. Goodison Park roared in disapproval but after the start to the season they have endured they treated this draw like a victory.
Goals: Beattie (pen, 37) 1-0; Lampard (50) 1-1.
Everton (4-4-2): Martyn; Hibbert, Yobo, Weir, Valente (Ferrari, 45); Kilbane, Cahill (Davies, 70), Neville, Arteta; Ferguson (Bent, 78), Beattie. Substitutes not used: Wright (gk), McFadden.
Chelsea (4-1-4-1): Cech; Gallas, Huth, Terry, Del Horno; Makelele; Wright-Phillips (Gudjohnsen, 58), Lampard, Essien, Cole (Robben, 66); Drogba (Crespo 71). Substitutes not used: Ferreira, Cudicini (gk).
Booked: Everton: Arteta, Ferrari. Chelsea: Drogba, Huth.
Referee: M Clattenberg (Tyne and Wear).
Man of the match: Beattie.
Attendance: 36,042.Reuse content