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.
“It was a unique day in a lot of ways,” Baines said. “It was just a little different. It is never nice when a manager loses his job but then everyone had to get on with their jobs as well.
“Winning a game is always a positive. I’m not too sure about what’s next. Things moved so quickly today.”
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.
“The FA Cup is really important for us. It’s a great competition, the only strange thing is to be playing on a Thursday night,” Fulham boss Marco Silva said.
