Match ends, Everton 2, Rotherham United 1.
Everton needed an extra-time winner from Abdoulaye Doucoure to finally see off Championship side Rotherham and reach round four of the FA Cup.
The Premier League side - featuring the likes of James Rodriguez and Lucas Digne - led through Cenk Tosun.
But Rotherham forced saves from Robin Olsen before Matthew Olosunde hooked in a deserved equaliser after the break.
A VAR call denied Tosun a second at the death but substitute Doucoure steered home a winner early in extra time.
It was tough on Rotherham, who would have earned a replay after it was level at the end of normal time were it not for the need for FA Cup ties to be resolved in one match this year - meaning no replays - as a result of Covid-19's impact on the football calendar.
On another day, a passive Everton display over the 90 minutes might have been enough to edge through such a tie but Rotherham clearly fancied the chance to reach round four for the first time in 18 years and inject some joy into a tough season battling Championship relegation.
After Tosun's clever chipped finish - his first Everton goal since 2019 - Paul Warne's side rallied, with Olosunde and Daniel Barlaser seeing shots saved, while Matt Crooks glanced a header over.
When Olosunde did level after the break, Everton manager Carlo Ancelotti's decision to not name top-scorer Dominic Calvert-Lewin or Brazilian forward Richarlison on the bench made the home side appear slightly vulnerable.
But he did introduce Bernard and Doucoure, who slotted into the bottom corner to keep Everton's hopes of landing a first major trophy since 1995 alive.
'Everton's tough day' - reaction
Everton assistant manager Duncan Ferguson: "It was a tough match so you are always fearing the worst when they got a corner in the last minute, so it was a wee bit of relief.
"We started well and Cenk took his goal well. It was a tough game for us and one we were delighted to get through."
Rotherham manager Paul Warne: "I thought we took the game to them after we conceded and I thought we were the better side in the first half.
"We could have got a 2-1 in and no-one would have complained about it."
Everton travel to Wolves on Tuesday while Rotherham have a week until they go to Derby on 16 January.
- 12DigneSubstituted forMinaat 66'minutes
- 26DaviesSubstituted forG Sigurdssonat 66'minutes
- 21André Gomes
- 17IwobiSubstituted forDoucouréat 61'minutes
- 19RodríguezSubstituted forNkounkouat 95'minutes
- 24GordonSubstituted forBernardat 61'minutes
- 10G Sigurdsson
- 22OlosundeSubstituted forJonesat 90'minutes
- 19HardingSubstituted forClarkeat 83'minutes
- 16LindsaySubstituted forJozefzoonat 76'minutes
- 25CrooksSubstituted forVassellat 67'minutes
- 24SmithSubstituted forHirstat 77'minutesBooked at 106mins
- Stuart Attwell
- Shots on Target
Second Half Extra Time ends, Everton 2, Rotherham United 1.
Foul by Abdoulaye Doucouré (Everton).
Daniel Barlaser (Rotherham United) wins a free kick on the left wing.
Gylfi Sigurdsson (Everton) wins a free kick in the defensive half.
Foul by Florian Jozefzoon (Rotherham United).
Attempt blocked. Kyle Vassell (Rotherham United) right footed shot from the right side of the box is blocked. Assisted by George Hirst.
Attempt blocked. Trevor Clarke (Rotherham United) left footed shot from the left side of the box is blocked. Assisted by Florian Jozefzoon with a cross.
Attempt blocked. Kyle Vassell (Rotherham United) right footed shot from the right side of the box is blocked. Assisted by Florian Jozefzoon.
George Hirst (Rotherham United) is shown the yellow card for a bad foul.
Yerry Mina (Everton) wins a free kick in the defensive half.
Foul by George Hirst (Rotherham United).
Second Half Extra Time begins Everton 2, Rotherham United 1.
First Half Extra Time ends, Everton 2, Rotherham United 1.
Foul by Séamus Coleman (Everton).
Trevor Clarke (Rotherham United) wins a free kick in the defensive half.
Niels Nkounkou (Everton) wins a free kick in the defensive half.
Foul by Michael Ihiekwe (Rotherham United).
Corner, Everton. Conceded by Richard Wood.
Attempt saved. Séamus Coleman (Everton) right footed shot from outside the box is saved in the bottom right corner. Assisted by Bernard.
- Stream eight live FA Cup third-round games this weekend on BBC iPlayer, the BBC Sport website and app. Find out more here.