Blackpool won 3-2 at Bloomfield Road after a 2-2 draw at the end of 120 minutes.
Sam Allardyce's West Brom, which is second-bottom of the Premier League, featured seven changes and it needed goals from Semi Ajayi and Matheus Pereira to send the tie to penalties.
In the shoot-out, Kyle Edwards, Darnell Furlong and Pereira all missed to hand the 1953 FA Cup winners a memorable victory.
After 21 winless games, Sheffield United finally earned a victory as Jayden Bogle's strike clinched a 3-2 success against third-tier Bristol Rovers.
Chris Wilder's 100th win as Blades boss was the first for the Premier League's bottom club since defeating Chelsea in July.
Leicester City rested Jamie Vardy and James Maddison at second-tier Stoke City but still won 4-0 as James Justin, Marc Albrighton, Ayoze Perez and Harvey Barnes all netted.
Everton needed extra-time to see off second-tier Rotherham United 2-1 at Goodison Park.
Cenk Tosun scored his first Everton goal since November 2019 in the first half and Abdoulaye Doucoure won it in the 93rd minute.
Top-flight struggler Fulham prevailed 2-0 after extra-time in its west London derby at Queens Park Rangers.
Burnley edged past third-tier Milton Keynes Dons 4-3 on penalties after Matej Vydra's stoppage-time equaliser for the Clarets rescued a 1-1 draw.
Monday's (AEDT) fixtures will sees eighth-tier Marine play at home to Jose Mourinho's Tottenham Hotspur, while Manchester City plays Birmingham City and Chelsea entertains Morecambe.