It was the ideal way to begin a year in which the Bluebirds, led by manager Malky Mackay, hope to secure Premier League football for the first time.
Joe Mason's 41st-minute strike made the difference at St Andrew's, with the former Plymouth striker turning the ball in after an error from home goalkeeper Jack Butland.
Hull stayed second following a goalless draw at Blackpool, after Middlesbrough stumbled to a 3-1 loss at Derby.
Michael Jacobs, Jeff Hendrick and Conor Sammon put Derby 3-0 ahead, with well-beaten Middlesbrough mustering a 90th-minute reply from Lukas Jutkiewicz.
Crystal Palace climbed above Middlesbrough into third, two points behind Hull, after their fine 3-1 home success over Wolves, while Leicester thrashed Huddersfield 6-1 to take a firm grip on fifth place.
Palace cruised to a 3-0 lead over a Wolves side lodged in lower mid-table, with Andre Moritz scoring after 31 and 52 minutes, either side of a Yannick Bolasie effort.
Wolves responded with a goal from Sylvan Ebanks-Blake 15 minutes from the end.
Leicester's new signing Chris Wood made an instant impact against Huddersfield after his arrival from West Brom, as the New Zealand striker - who had a productive loan spell with Millwall in the first half of the season - scored twice in the opening 24 minutes.
Ritchie de Laet made it 3-0 before half-time before French midfielder Anthony Knockaert scored a quickfire pair of goals soon after the break.
Scott Arfield pulled a goal back for Huddersfield, but Martyn Waghorn had the final say for the Foxes with their sixth goal in the 76th minute.
Blackburn caretaker manager Gary Bowyer oversaw a 3-0 home win over Nottingham Forest, the goals coming from Ruben Rochina, Jordan Rhodes and Colin Kazim-Richards.
And Brighton won by the same margin against Ipswich, the points at Portman Road going their way after Dean Hammond, Craig Mackail-Smith and Wayne Bridge - from 25 yards - netted.
Leeds edged out Bolton 1-0 at Elland Road with a penalty from Luciano Becchio after 66 minutes, while Millwall were 2-1 victors against Bristol City on the back of goals from Dany N'Guessan and Adam Smith. Jon Stead was Bristol City's scorer.
Peterborough climbed out of the relegation zone as their improvement continued with a 2-1 home win against bottom side Barnsley.
Michael Bostwick put Peterborough ahead after 28 minutes, and although they conceded an equaliser to Marlon Harewood in the 79th minute, Tommy Rowe fired in late on at London Road to earn the points.
Sheffield Wednesday slid below Peterborough and into the bottom three after second-half goals from Keith Treacy and Ross Wallace - a penalty - brought Burnley a 2-0 victory at Hillsborough.
Watford slumped to a 4-3 home defeat in the lunchtime clash with Charlton at Vicarage Road.
Charlton captain Johnnie Jackson headed a 78th-minute clincher to secure a first win in seven for Chris Powell's side.
Daniel Pudil, Almen Abdi and Alexandre Geijo had earlier scored for Watford, with Charlton replying through Tommie Hoban's own goal and two goals from Yann Kermorgant.