Match ends, Everton 3, Rotherham United 1.
Everton continued their unbeaten start to the season with a victory over Rotherham that secures their place in the third round of the Carabao Cup.
Despite making eight changes from Saturday's Premier League draw at Bournemouth, the Toffees had too much for their Championship opponents.
Gylfi Sigurdsson opened the scoring with a neat close-range finish.
Two goals from Dominic Calvert-Lewin, either side of Will Vaulks' header for Rotherham, saw the home side through.
The win extends Everton's League Cup record of having progressed from the second round in every season since 2001-02, when they were beaten on penalties by Crystal Palace.
Added to the two draws and a win from their first three Premier League games, it makes for a very encouraging start to the reign of boss Marco Silva.
Silva will have been pleased with the display of some of his second string, a number of whom pressed their case for inclusion for Saturday's home league encounter with Huddersfield.
Sandro Ramirez provided a neat low cross for Sigurdsson to score the opener, while Lucas Digne, at left-back, and fellow full debutant Kurt Zouma, at centre-back, were assured.
Calvert-Lewin was the standout, though, rising high in the box to head in Digne's pinpoint cross and then curling in a superb second from 20 yards just moments after Vaulks had given Rotherham hope with his headed goal.
It was a timely reminder of the young forward's goalscoring ability with current leading scorer Richarlison suspended at the weekend following his red card at the Vitality Stadium.
Everton boss Marco Silva: "It was important for us to get the result. That was our goal before the match, of course. But the first words I must give to the fans. It was fantastic support from them tonight.
"It's a Wednesday evening, our first League Cup match and more than 31,000 came to support us. That is fantastic for us.
"They showed me and our players that they are behind us and supporting us, that they believe in our squad.
"For me, that is just as important as the result for us tonight."
Rotherham boss Paul Warne: "I don't look at my players as first-team players or reserves, they are much of a muchness.
"I don't regard making all those changes as weakening the team. It isn't like I've been offensive to the cup, I've got another big game Saturday.
"I was intending to throw the kitchen sink at them in the last 20 minutes and, at 2-1, it looked like I was a fantastic tactician."
- 12DigneBooked at 52mins
- 9RamírezSubstituted forSchneiderlinat 86'minutes
- 10G SigurdssonSubstituted forWalcottat 66'minutes
- 19NiasseSubstituted forTosunat 74'minutes
- 33Neves Virgínia
- 25WilesSubstituted forVaulksat 45'minutes
- 22NewellSubstituted forTaylorat 72'minutes
- 10BallSubstituted forVassellat 61'minutes
- Graham Scott
- Shots on Target
Second Half ends, Everton 3, Rotherham United 1.
Attempt missed. Kieran Dowell (Everton) right footed shot from the centre of the box misses to the right. Assisted by Lucas Digne.
Attempt saved. Sean Raggett (Rotherham United) header from the centre of the box is saved in the bottom right corner. Assisted by Anthony Forde with a cross.
Offside, Everton. Dominic Calvert-Lewin tries a through ball, but Cenk Tosun is caught offside.
Attempt saved. Cenk Tosun (Everton) right footed shot from outside the box is saved in the top centre of the goal. Assisted by Kieran Dowell.
Attempt blocked. Kyle Vassell (Rotherham United) left footed shot from outside the box is blocked. Assisted by Jon Taylor.
Corner, Everton. Conceded by Anthony Forde.
Theo Walcott (Everton) wins a free kick in the defensive half.
Foul by Jamie Proctor (Rotherham United).
Goal! Everton 3, Rotherham United 1. Dominic Calvert-Lewin (Everton) right footed shot from the left side of the box to the top right corner. Assisted by Tom Davies.
Substitution, Everton. Morgan Schneiderlin replaces Sandro Ramírez.
Goal! Everton 2, Rotherham United 1. Will Vaulks (Rotherham United) header from the centre of the box to the top right corner. Assisted by Anthony Forde with a cross following a corner.
Corner, Rotherham United. Conceded by Maarten Stekelenburg.
Attempt saved. Jon Taylor (Rotherham United) right footed shot from outside the box is saved in the bottom left corner. Assisted by Will Vaulks with a headed pass.
Offside, Everton. Sandro Ramírez tries a through ball, but Cenk Tosun is caught offside.
Foul by Dominic Calvert-Lewin (Everton).
Semi Ajayi (Rotherham United) wins a free kick in the defensive half.
Attempt missed. Jamie Proctor (Rotherham United) right footed shot from the centre of the box is too high. Assisted by Semi Ajayi with a headed pass.
Corner, Rotherham United. Conceded by Mason Holgate.