Dyche was axed less than four hours before the third-round tie, with the club's new owners, the Friedkin Group, reportedly in talks with former Everton boss David Moyes about a return to Goodison Park.
Moyes is the bookmakers' favourite for the job of saving Everton from relegation, 12 years after he left for Manchester United.
The Scot, who managed Everton in a successful spell between 2002 and 2013, has been out of work since leaving West Ham at the end of last season.
The Toffees are once again battling to avoid relegation after winning just one of their past 11 games, with that sole success coming against lowly Wolves in December.
Dyche was at the Everton training ground earlier on Thursday to say his goodbyes after the axe fell, while the matchday programme for the Peterborough game still contained his column.
Former Everton defender Leighton Baines, now the club's under-18s coach, and captain Seamus Coleman took charge against Peterborough, using the starting lineup selected by Dyche before his dismissal.
Peterborough's Cian Hayes' was inches away from giving them a shock first-half lead when his low drive was cleared off the line by Jarrad Branthwaite.
But Beto finally lifted the mood among Everton's subdued fans in the 42nd minute as he took Harrison Armstrong's pass inside the area, rounded Peterborough keeper Nicholas Bilokapic and slotted into the empty net.
In a blow to Everton, on-loan Chelsea striker Armando Broja needed oxygen before being stretchered off with his right ankle heavily strapped.
There was no family affair for Everton defender Ashley Young, who came off the bench in the closing minutes, while his teenage son Tyler, who plays for Peterborough, remained an unused substitute.
But Branthwaite was fouled by Jadel Katongo in stoppage time and Iliman Ndiaye converted the penalty as Everton made a winning start to life without Dyche.
Fulham powered to a 4-1 win over second-tier Watford at Craven Cottage.
The Premier League side took the lead through Rodrigo Muniz's instinctive strike from Adama Traore's cross in the 26th minute.
Rocco Vata levelled for Watford with a superb blast from 25 yards in the 33rd minute.
Raul Jimenez's 49th-minute penalty - awarded for Ryan Porteous's foul on Traore - restored Fulham's advantage.
Joachim Andersen's 65th-minute volley effectively sealed the victory before Timothy Castagne headed in with five minutes left.
Cardiff knocked out promotion-chasing Sheffield United with a 1-0 win at Bramall Lane.
United are 20 places above struggling Cardiff in the Championship and beat them 2-0 in December, but Cian Ashford's 19th-minute goal sent the Bluebirds into the last 32.