The former Manchester City man has largely found himself out of Roberto Di Matteo's plans this season, but recently netted against Tottenham in the league before starting, and scoring, against United.
Sturridge attributed an early miss when on goal to rustiness due to a lack of action, and is hoping to have done enough to seal a spot in his favoured central striking role.
"I've been out of any sort starting positions and any substantial minutes so I didn't feel as sharp in front of goal," he told Sky Sports.
"When you train you can score as many goals but it's completely different in a game. I'm delighted I scored but I know I need to get sharper.
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|9/2||Daniel Sturridge is 9/2 with William Hill to open the scoring at Swansea
"I'm really happy the manager gave me an opportunity up front today because that's what I've wanted throughout my Chelsea career.
"That's probably one of five starts I've had as striker so I'm just really happy he's put faith in me tonight and hopefully he'll do it again in future."
The striker hailed the crucial nature of the win to get Chelsea supporters back on side after losing to the Red Devils in Sunday's Premier League.
"It's important we came out here with the right attitude because in the previous game the manager emphasised that we didn't start the game well enough," Sturridge added.
"We just wanted to put a smile on the fans' faces again as they were a bit disappointed after the weekend.
"It was great to play in - I watched the Arsenal game last night and it was hard to match that one but I feel we got close to it.
"It's always good to not get a difficult side but in this competition it's just one game and anything's possible so it doesn't really matter who we play, we just have to have the right mentality and we could be OK for it."
Victor Moses also impressed for the home side and won the first of Chelsea's two penalties, which David Luiz converted, but was quick to heap praise on his team-mate Sturridge.
"We gave a silly first goal away but we got back and Sturridge was outstanding today, as were the rest of the boys," Moses said.
"After the first half we came out and dominated the game so we were very delighted with the win."