The Blues were on the wrong end of a controversial Premier League defeat against the Red Devils on Sunday, with Javier Hernandez scoring a late winner and referee Mark Clattenburg sending two Chelsea players off before being accused of using "inappropriate language" to a player.
But the Londoners fought back from 2-0 down to make it 3-3 at the end of normal time in their midweek rematch at Stamford Bridge, with further goals from Sturridge and Ramires enough to cancel out Ryan Giggs' late penalty.
Sturridge told Sky Sports: "It's great. It's important that we came out here with the right attitude because the manager criticised us for not starting well enough (on Sunday).
"It's just about putting a smile on the fans' faces again because I feel that they were disappointed after the weekend."
It was Sturridge's first start of the season with boss Roberto Di Matteo electing to start Fernando Torres in every game before Wednesday's clash.
"I've been out of the starting positions so I didn't feel as sharp in front of goal," he said.
"I'm delighted I scored but I know I need to get sharper.
"I'm really glad the manager gave me the chance up front today, that's all I've wanted throughout my Chelsea career.
"I'm really happy that he's put faith in me tonight and hopefully he'll do it again."