The Tigers required penalties to progress 5-4 in Monday night's (Tuesday AEST) shootout after Kieran Dowell cancelled out Robert Bozenik's goal in a 1-1 draw.
Herrington had come off the bench at the weekend to make his Premier League debut in a shock 2-0 home win over Manchester United.
Unlike at the World Cup where Herrington missed a spot kick in Australia's round-of-32 loss to Egypt, the teenager was not called upon at Bet365 Stadium against the second-tier Potters.
Leeds United secured a second win at Premier League rivals Nottingham Forest in three days to book a spot in the third round.
Dan James' 50th-minute opener and a Neco Williams own goal earned a 2-0 victory for Daniel Farke's side to back up Saturday's 1-0 top-flight success.
Before the weekend, the West Yorkshire club had not won at the City Ground since 1972.
Callum Wilson scored just six minutes into his full debut and captain Nathan Collins headed home twice as Brentford hammered Birmingham City 6-1 at St Andrew's.
Premier League new boys Coventry City survived a scare to beat League One Plymouth Argyle 4-2.
Marcelino Nunez scored a sensational goal from the halfway line as part of a brace to help Ipswich Town beat Leicester City 3-1.
League One Peterborough United stormed into round three by thumping Championship Watford 5-1 at Vicarage Road, while West Ham United came from behind to thrash Championship rivals Southampton 4-1.
Middlesbrough avenged last season's exit at the hands of Doncaster Rovers with a 3-1 win, Norwich City enjoyed a 2-1 victory in Cardiff and Wolverhampton Wanderers beat Sheffield Wednesday 2-0.
Elsewhere, Fleetwood Town thrashed League Two rivals Shrewsbury Town 4-0, Sheffield United secured a 2-1 victory at fellow Championship side Blackburn Rovers and Leyton Orient won 6-5 on penalties away to Walsall following a goalless draw.
Lincoln City were 4-1 victors at Blackpool, Millwall won 1-0 at Cambridge United, Barnsley overcame fourth-tier Crewe Alexandra 3-2 and Reading eased to a 3-0 win at Stevenage.
With AAP.
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