The Bluebirds, who were promoted from the Championship last season, fell to a 3-1 home defeat at the hands of second-tier side Norwich City on Tuesday night.
Crystal Palace avoided a potential banana skin by winning 1-0 at Swansea City, while Leicester City made light work of Fleetwood Town with a 4-0 win at the King Power Stadium and Bournemouth ran out 3-0 winners over MK Dons.
Southampton emerged triumphant from their south coast derby, winning 1-0 at Brighton & Hove Albion, while West Ham United, Fulham and Wolverhampton Wanderers also made it into the third round with victories over AFC Wimbledon, Exeter City and Sheffield Wednesday respectively.
Blackburn Rovers are among the Championship sides to progress as they ran out 4-1 winners over Lincoln City, with Brentford also through thanks to a 1-0 win over Cheltenham Town. Derby County, Middlesbrough, Queens Park Rangers and West Bromwich Albion join their fellow second-tier sides in the hat.
Elsewhere, Blackpool ran out 2-1 winners over Doncaster Rovers, Oxford United thrashed Newport County 3-0 at Rodney Parade and Wycombe Wanderers overcame Forest Green Rovers on penalties following a 2-2 draw.
Results in full:
Swansea City 0-1 Crystal Palace, AFC Wimbledon 1-3 West Ham United, Blackburn Rovers 4-1 Lincoln City, Bournemouth 3-0 MK Dons, Brentford 1-0 Cheltenham Town, Brighton & Hove Albion0-1 Southampton, Burton Albion 1-0 Aston Villa, Cardiff City 1-3 Norwich City, Doncaster Rovers 1-2 Blackpool, Fulham 2-0 Exeter City, Hull City 0-4 Derby County, Leeds United 0-2 Preston North End, Leicester City 4-0 Fleetwood Town, Middlesbrough 2-1 Rochdale, Newport County 0-3 Oxford United, Queens Park Rangers 3-1 Bristol Rovers, Sheffield Wednesday 0-2 Wolverhampton Wanderers, Walsall 3-3 Macclesfield Town (1-3 on penalties), Wycombe Wanderers 2-2 Forest Green Rovers (4-3 on penalties), Stoke City 2-0 Huddersfield Town, West Bromwich Albion 2-1 Mansfield Town