The England international, who completed his £12m move from Chelsea to Anfield on Friday, scored from close range in the seventh minute to make the perfect start to his Liverpool career.
Sturridge was replaced by Luis Suárez in the second half, and the Uruguay international netted his 19th goal of the season, albeit with a strong hint of handball.
Mansfield netted a late consolation through Matt Green, but Brendan Rodgers’ side clung on to book a fourth-round meeting with Oldham Athletic.
“I enjoyed it [his debut] a lot,” Sturridge told ESPN. “It was my first game for the team and it was great to get to know my team-mates on the field, and we won the game.
“It was a great ball from [Jonjo] Shelvey and I got on the end of it and put it away. But the team played fantastic and we showed a lot of character, so we’re happy to get the win.
“I know [I could have had a hat-trick]. I’m disappointed but the team won so that’s what matters.”
“Wherever the manager plays me, I’ll be on the pitch and able to play there so it doesn’t really matter what position. He signed me as a centre forward but wherever I’m asked to play, I’ll be delighted to do so.”
Liverpool’s Jamie Carragher, who captained the Reds in the absence of Steven Gerrard, heaped praise on Sturridge and backed the new signing to add more firepower to their frontline.
“It was a great debut for Daniel. It was a great goal and you could see his class when we got the ball into him. He had a bit of pace so we could hit them in behind,” Rodgers added.
“But we’re playing non-league opposition so you’d expect that. But I’m sure he can do that against Premier League defences, so that’s what we want to see.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news