Ian Holloway's men were the only top-flight casualties in Tuesday's action but the Cottagers had to come back against Burton Albion and Sunderland overturned a two-goal deficit
Fulham were pushed all the way by League Two side Burton Albion, with Jack Dyer and Michael Symes putting the hosts ahead after Adel Taarabt had scored his first goal for the Cottagers.
However, Hugo Rodallega equalised three minutes from the end of extra time before Martin Jol's men won through 5-4 on penalties.
Sunderland were almost on the end of a giant-killing against League One's MK Dons, as they came from two goals down to edge into the third round 4-2.
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Hull City Tigers
Fellow Premier League sides Southampton and West Brom enjoyed a more comfortable evening as they dispatched Barnsley and Newport County, respectively.
Mauricio Pochettino's men were 5-1 winners at Oakwell, while Saido Berahino's hat-trick ensured progression for West Brom at the Hawthorns in a 3-0 victory.
Norwich City ran out 6-3 winners against Bury in a nine-goal thriller, with Johan Elmander getting his first two goals for the club, while West Ham saw off Cheltenham Town 2-1.
Hull City Tigers edged through against League One leaders Leyton Orient thanks to a Robert Brady goal in extra time.
Reading were on the wrong end of a 6-0 thumping against Peterborough United while goals from Nile Ranger and Alex Pritchard gave Swindon Town a 2-0 victory over QPR at Loftus Road.
Tranmere Rovers dumped out Bolton Wanderers at Prenton Park on penalties, Jermaine Beckford having equalised after Cole Stockton had given Tranmere the lead. However, Ronnie Moore's men kept their cool from the spot to win 4-2 in the shootout.
The all-Championship clash between Yeovil Town and Birmingham City saw Lee Clark's men progress on penalties while two late goals from Jonathan Hogg and Adam Hammill gave Huddersfield Town a slender 3-2 victory over Charlton.
Two first-half goals gave Burnley bragging rights in the Lancashire derby against Preston North End while Leeds United overcame Yorkshire neighbours Doncaster Rovers 3-1.
Derby County and Leicester City are both into the next round, as well, after seeing off Brentford and Carlisle United, respectively.