Everton 2 Dagenham and Redbridge 0: Kone and Mirallas send Martinez's men through
Arouna Kone's header and Kevin Mirallas' late penalty were enough to give Everton a 2-0 win at home to League Two side Dagenham and Redbridge in the FA Cup third round.
Kone gave Everton a first-half lead with a fine header from Bryan Oviedo's free-kick, while visiting goalkeeper Mark Cousins made a string of saves to keep his team in the match.
Cousins denied Mirallas and Ramiro Funes Mori in the first half and frustrated Kone and Oviedo after the break.
Dagenham struggled to create any meaningful chances on the club's first appearance in the third round since 2012.
Mirallas converted from the spot after being fouled in the box to ensure Everton's name will be in the hat for the fourth-round draw on Monday.
Everton boss Roberto Martinez handed rare starts to Oviedo, Mirallas and Darron Gibson, while captain Phil Jagielka returned after two months out with a knee injury.
Steven Pienaar, making his first appearance for Everton since April, should have given the hosts the lead after just three minutes.
Oviedo got down the right wing and delivered a fine cross, but the South Africa winger's header flew over the crossbar.
Mirallas then tested Cousins before Aaron Lennon failed to pick out Kone from a good position.
Cousins was in action again after 24 minutes, producing a fine save from Funes Mori's 25-yard curler.
Everton were dominating the ball without much penetration, but took the lead after 31 minutes. Oviedo floated a free-kick to the back post and Kone headed the ball across Cousins and in.
Funes Mori was substituted shortly before the break after sustaining a nasty forehead cut, with Matthew Pennington coming on to make his second appearance for Everton.
As torrential rain fell at Goodison Park, Kone almost made it 2-0 on the hour mark. The striker hit a low shot towards the near post, but the impressive Cousins was again equal to the effort.
Muhamed Besic then ran the length of the pitch to set up Kone, but the Ivory Coast attacker could not double his tally for the afternoon.
Dagenham manager John Still, appointed on New Year's Eve to begin a third spell in charge of the club, introduced 40-year-old striker Jamie Cureton from the bench in a bid to force an equaliser.
Oviedo let fly from range on 72 minutes and Cousins spilled the shot, but Brendan Galloway could not turn the rebound into the net, before Kone planted a header just wide from a good position.
Mirallas then secured the win from 12 yards after being brought down in the box with just six minutes to go.
Everton closed out the match comfortably - a cat making a late appearance on the pitch to briefly halt play - to record back-to-back cup victories after Martinez's men beat Manchester City 2-1 in the League Cup semi-final first leg on Wednesday.