A goal four minutes from the break from Beto and a penalty deep in stoppage time from Iliman Nydiaye secured a 2-0 win for Everton.
In cup-ties, often it comes down to the big chances and Peterborough United will feel they had a couple of notable opportunities in front of goal at Goodison Park, but the Premier League quality eventually shone through as Everton claimed a 2-0 victory to progress in the Emirates FA Cup.
Posh defended resolutely for long spells against the Toffees who parted company with manager Sean Dyche ahead of kick-off but were undone by a lapse in concentration before the break and a late penalty kick.
Nick Bilokapic kept out Beto's header and Gueye's deflected hit from the edge of the box, but otherwise, was largely untroubled.
Ricky-Jade Jones was denied by a last ditch tackle in the first half after an excellent move involving Cian Hayes and Chris Conn-Clarke.
From a set-piece, the ball fell to Hayes at the back post, but his volley was cleared off the line by Brathwaite.
The opening goal arrived on 41 minutes as BETO beat the offside trap, rounded the goalkeeper Bilokapic and slotted into an empty net.
Posh made changes to get back into the game, two at the break and further substitutions in the second half and Hayes put a number of delicious crosses into the box, but there were no takers.
Gustav Lindgren found the side-netting with an effort after a strong run but after 10 minutes of injury time, most for an injury to Broja, the hosts found a second goal as NDIAYE slotted home from the spot after Katongo was penalised for a foul at the back post.
Everton: Virginia, Patterson (sub Young 72min), Keane, Mangala (sub Lindstrom 88min), Ndiaye, Beto (sub Broja 67min, sub Doucare 88min)), O’Brien, Mykolenko, Gueye, Branthwaite, Armstrong (sub Harrison 67min). Unused subs: Begovic, Leban, Butterfield.
Posh: Bilokapic, Dornelly (sub Odoh 76min), Nevett (sub Wallin 46min), Fernandez, Hayes, O'Brien-Brady (sub Mothersille 87min), Kyprianou (sub De Havilland 63min), Katongo, Collins, Conn-Clarke, Jones (sub Lindgren 46min). Unused subs: Blackmore, Young, Ihionvien, Sparkes.
Half-time: Everton 1-0 Peterborough United
Full-time: Everton 2-0 Peterborough United
Attendance: 38,095
Referee: Thomas Bramall