Serbia striker Aleksandar Mitrovic fired home his ninth goal in nine games 12 minutes from time at Hillsborough to stretch the Whites' unbeaten run to 19 matches.
The Bluebirds have a game in hand but they have to respond to Friday night's 1-0 home defeat against Wolverhampton Wanderers.
Derby County sit fifth on 68 points with a game in hand on their rivals and were triumphant on Saturday as they saw off Bolton Wanderers 3-0 at Pride Park.
Sixth-placed Middlesbrough, meanwhile, ran out 2-0 winners over Nottingham Forest to remain in contention for a playoff spot.
Millwall's superb form continued with a 2-0 win over Bristol City, putting the Lions seventh in the table and just one point behind Boro.
At the opposite end of the table, Barnsley beat Sheffield United 3-2 to give their survival hopes a huge boost, the Tykes now two points behind 21st-placed Bolton and with a game in hand.
Sunderland, meanwhile, could only claim a 1-1 draw at Leeds United, the same scoreline as in the relegation six-pointer between Birmingham City and Burton Albion.
Hull City's renaissance continued with a 4-0 win over Queens Park Rangers, while Reading secured a crucial 1-0 victory against Preston North End.
Elsewhere, Brentford secured a 1-0 home win over Ipswich Town, Neal Maupay converting a second-half penalty to seal all three points at Griffin Park.
Results in full: Barnsley 3-2 Sheffield United, Birmingham City 1-1 Burton Albion, Brentford 1-0 Ipswich Town, Derby County 3-0 Bolton Wanderers, Hull City 4-0 Queens Park Rangers, Leeds United 1-1 Sunderland, Middlesbrough 2-0 Nottingham Forest, Millwall 2-0 Bristol City, Reading 1-0 Preston North End, Sheffield Wednesday 0-1 Fulham, Norwich City 3-1 Aston Villa