The visitors fell behind after 19 minutes at Old Trafford when Patrice Evra’s loss cross was turned past Pepe Reina by Robin van Persie for the Dutch striker’s 17th goal of the season.
Sir Alex Ferguson’s side doubled their advantage in the 54th minute, with Van Persie’s free-kick eventually finding the net courtesy of Nemanja Vidic’s inadvertent header.
Rodgers replaced Lucas Leiva with new signing Sturridge at half-time, and the former Chelsea man scored from close range shortly after United’s second, before missing a late chance to level.
“In the second half, after we conceded the goal, we could have given up the fight but that is nature of the group,” Rodgers told BBC Sport.
“In the end, I thought we maybe deserved something from the game. The goals we conceded were disappointing. We will learn from days like today, even though we didn’t get the result.
“In the first half we never threatened enough. There wasn’t too many chances either way and the ones they had we presented to them.”
The Reds boss added: “He [Daniel Sturridge] was terrific when he came on and we were much better in the second half.”
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