Abdoulaye Doucoure emerged from the bench to spare Everton’s blushes in their FA Cup third-round tie against Rotherham at Goodison Park, winning 2-1.
The Toffees took the lead when Cenk Tosun dinked home with eight minutes on the clock.
However, Matthew Olosunde drew Paul Warne’s side level just after the break when he reacted quickest to the break of the ball in the box; neither side could prise a winner in the regulation 90, sending the tie to extra-time.
And Doucoure, a 61st-minute substitute for Alex Iwobi, latched on to a James Rodriguez through-ball to slot past Jamal Blackman just three minutes into extra-time to seal the win for Everton.
Norwich City strolled to a 2-0 win over fellow Championship (second-tier) side Coventry City after midfielder Kenny McLean and forward Jordan Hugill scored inside seven minutes.
Millwall's Danish striker Kenneth Zohore and defender Shaun Hutchinson led the Championship side to a 2-0 win over fifth-tier third-round debutants Boreham Wood, while Nottingham Forest beat Cardiff City 1-0 with an early goal by striker Lyle Taylor.
Luton Town beat Reading 1-0 in another all-Championship clash to reach the fourth round for the first time since the 2012-13 season thanks to midfielder George Moncur.
Plymouth Argyle ran out 3-2 winners as they visited Huddersfield, while Brentford defeated fellow Championship side Middlesbrough.
Leicester City cruised to a comfortable 4-0 win over Stoke City at the Potters' ground, and Sheffield Wednesday beat Exeter City 2-0. Another South West side Bristol Rovers could not quite take advantage of Sheffield United's poor Premier League form with the Blades winning 3-2.
Former Premier League side Bournemouth trounced Oldham Athletic 4-1, Swansea City beat Stevenage 2-0, Burnley needed penalties (4-3) to get past MK Dons, and Wycombe defeated Preston 4-1.
Fulham beat West London rivals QPR 2-0 at Loftus road, Blackburn Rovers fell 1-0 at home to Doncaster Rovers, and West Brom's wretched season continued as they lost on penalties (3-2) to Blackpool.