Steve McClaren's men lost 2-1 at home to Reading on Saturday thanks to a late strike courtesy of Yakubu, meaning Derby missed out on a quarter-final place.
But Derby are enjoying a fine league campaign and head into the trip to Rotherham just a point behind pacesetters Bournemouth.
And Lingard, a January arrival on loan from Manchester United, has urged his new team-mates to produce a response to keep Derby in the mix at the top of the Championship.
He told the club's official website: "We need to move on because we have got 16 cup finals now until the end of the season.
"Every game is going to be so important now. We have got a lot of Saturday, Tuesday games and we are going to be fully focused to try and pick up as many points as possible over the remaining 16 games of the season."
While Derby will be looking to forget FA Cup disappointment, Blackburn Rovers will be hoping their 4-1 hammering of Premier League side Stoke City provides momentum ahead of a visit to Cardiff City, who have not won in five Championship games.
And the other midweek Championship contests see struggling Wigan Athletic head to Reading needing a boost in their bid to avoid relegation, while Birmingham City welcome Middlesbrough on Wednesday.
Bradford City's battle to secure a League One play-off place continues at Leyton Orient on Wednesday, with Phil Parkinson's men going into the clash on the back of a 2-0 cup triumph over Sunderland.
"I am obviously delighted with the way we played," he told BBC Sport after the win.
"We did the right things in the right areas of the pitch. Defensively we were outstanding."
League One leaders Bristol City are in action on Tuesday as they host Peterborough United, while MK Dons could move up to second with a win at relegation battlers Colchester United.
Crewe Alexandra, Sheffield United and Chesterfield also face midweek trips, playing Doncaster Rovers, Notts County and Scunthorpe United respectively.
And in League Two's solitary fixture, Mansfield Town entertain promotion hopefuls Luton Town.
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