Everton looked like they could suffer an embarrassing Capital One Cup exit to League One side Stevenage but secured a 2-1 victory after extra-time.
Luke Freeman put the visitors ahead in the 36th minute in the second-round tie, but on-loan Barcelona winger Gerard Deulofeu managed to level on the stroke of half-time.
The Toffees were unable to stop the match going to extra-time, though, and it was not until Marouane Fellaini's goal five minutes from time that eventually Everton claimed their first win under Roberto Martinez.
Brothers Shola and Sammy Ameobi saved Newcastle's blushes a late 2-0 win at Morecambe.
Shola came off the bench to open the scoring with a deflected 84th-minute shot, with Sammy making sure of the Magpies' progress in the fourth minute of stoppage time.
It was a first win of the season for manager Alan Pardew, whose much-changed side looked unconvincing in setting up a home tie with Leeds.
Elsewhere, Kenwyne Jones-inspired Stoke saw off League One side Walsall 3-1 at the Britannia Stadium.
The Trinidad and Tobago striker put the hosts 2-0 ahead before Ashley Hemmings pulled one back in the 57th minute. There was to be no shock, though, with Jones netting a further goal to secure his hat-trick.
Aston Villa went some way to laying to rest the ghosts of last season's semi-final exit to Bradford by easing past Rotherham 3-0.
Andreas Weimann opened the scoring with a fine 20-yard effort at Villa Park, where Christian Benteke and Fabian Delph also netted.
Nicky Maynard and Rudy Gestede, who had earlier come on for the injured Andreas Cornelius, struck to ensure 2012 finalists Cardiff a 2-0 win at Accrington.
An Elliott Ward own goal and Cristian Battocchio chip saw Watford past Bournemouth 2-0 at Vicarage Road.
Nottingham Forest beat Millwall 2-1 after extra-time at the City Ground.
Liam Feeney levelled Matt Derbyshire's goal late in normal time, only for Jamaal Lascelles to head home the decider.
Information from the Press Association was used in this report.