Lambert had said earlier this week the cup was something he could do without and his side fell behind at Villa Park thanks to Jamie Murphy's deflected strike for the Blades.
Nicklas Helenius levelled for Villa, but Ryan Flynn's strike gave United a 2-1 victory and a place in the fourth round.
Cardiff manager Ole Gunnar Solskjaer made a winning start to his reign as his side came from behind to win 2-1 at Newcastle thanks to goals from Craig Noone and Fraizer Campbell.
Everton romped to a 4-0 win over QPR as Ross Barkley, Nikica Jelavic (2) and Seamus Coleman got the goals, with Jelavic also missing a penalty.
Southampton came out 4-3 winners in a seven-goal thriller against Championship high-fliers Burnley.
In the day's early kick-off Blackburn earned themselves a lucrative replay at the Etihad Stadium as they drew 1-1 with Manchester City. Alvaro Negredo had tapped in to give City the lead, but Scott Danns capitalised on Costel Pantilimon's error to level.
Dwight Gayle and Marouane Chamakh got the goals as Crystal Palace won 2-0 at West Brom while strikes from Kenwyne Jones and Charlie Adam helped Stoke to a 2-1 win over Leicester.
Norwich and Fulham will do it all again after a 1-1 draw at Carrow Road. Cup holders Wigan also face a replay at MK Dons after a thrilling 3-3 draw at the DW Stadium.