Everton set up a mouth-watering FA Cup semi-final against Merseyside rivals Liverpool with a 2-0 win over Sunderland on Tuesday night.
A fine first-half performance from the Toffees at the Stadium of Light was capped by a fine finish from Nikica Jelavic after good work from Magaye Gueye.
With Moyes' men dominating, O'Neill introduced David Vaughan in an attempt to strengthen the Black Cats' midfield, but the move back-fired horribly.
After Marouane Fellaini had played a ball through to the unmarked Jelavic, the Croatia international slid it under Mignolet, only for the retreating Vaughan to bundle the ball over his own goalline in comical fashion.
Liverpool await Moyes' men in the semi-final, while Sunderland will now look to concentrate on the chase for European football.
Tottenham booked their spot in the last four of the FA Cup with a dominant 3-1 win over Bolton in Tuesday's re-arranged quarter-final clash.
The match was originally first played 10 days ago, but had to be abandoned after 23-year old Bolton midfielder Fabrice Muamba collapsed on the White Hart Lane pitch after suffering from a cardiac arrest.
The deadlock was broken in the 74th minute when Ryan Nelsen nodded in a Rafael Van Der Vaart corner in a crowded penalty area.
Gareth Bale, who remained lively throughout the match, extended their lead just three minutes later after latching onto substitute Jermaine Defoe's through-ball and finishing with aplomb after cutting back onto his left foot.
Bolton, out-played, out-fought and out-thought for much of the match, managed to hit back in the 89th minute through Kevin Davies' tapped finish at the back post after an excellent Ivan Klasnic cross deceived the Tottenham backline.
Substitute Louis Saha scored the final goal of the game in the last minute of injury-time, with an effort from the edge of the area after spinning off his marker Zat Knight.
Bolton will be sweating on a potentially serious knee injury to Darren Pratley, who was stretchered off on 25 minutes after fouling Bale as he surged past him just inside the Bolton half.
