The Reds moved up to second in the standings after a 3-1 win at managerless Sunderland, with Sturridge opening the scoring before two goals from Luis Suarez - either side of Emanuele Giaccherini's effort for the Black Cats - sealing victory.
It was the Merseysiders' fourth win in six league matches, but Sturridge played down their hopes of challenging for the title.
"It's early days, take every day as it comes. Don't worry about the future, worry about the here and now," he said on Sky Sports 1.
"I don't think we played as well as we could have in the first half, even though we were 2-0 up, but it was very difficult in the second half.
"We took our foot off the pedal a little bit and once they scored a goal it got a bit more difficult for us but we held on in the end.
"We defended well towards the end of the game and then got a late goal, which killed it off."
Sturridge opened the scoring in controversial fashion, with the ball taking a heavy deflection off his arm before nestling in the net.
The England man defended his actions, however, adding: "It's a great ball from Stevie, it's gone over the defender's head and it was last minute, I've obviously tried to head it but I guess my heading's not good enough.
"The ball's just gone in off my arm but I wouldn't do that on purpose. Sometimes these things happen."
Suarez netted a brace on his first Premier League outing since his 10-game ban and Sturridge believes the Uruguayan's presence can only help Liverpool this season.
"It's great to have him back in, he gives us options and he's one of the best forwards in the league. We form a good partnership," said the former Manchester City and Chelsea striker.
"We miss Philippe Coutinho, with his injury, we've got some great players in this team and I'm happy to be playing with them."
Liverpool boss Brendan Rodgers was also delighted to have Suarez back in the fold.
The 26-year-old made his competitive return in the 1-0 Capital One Cup defeat at Manchester United in midweek, playing the entire 90 minutes.
While he drew a blank at Old Trafford, Suarez was a constant thorn in United's side and underlined his importance to Liverpool's cause with his brace against Sunderland.
"He's getting the reward for his hard work over the period that he was off. I thought he was excellent the other night at Old Trafford," Rodgers said on Sky Sports 1.
"Him and Daniel up front are a real, real handful. They combined very well."
The Northern Irishman was pleased Liverpool were able to bounce back from their defeat to David Moyes' men.
"On the back of a really tough game where we played well during the week, today was always going to be physically demanding game for us," he added.
"I thought the work we put into it was exceptional and to get the three goals and maybe a few more chances to score a few more.
"I thought we were really dangerous on the counter-attack and as a team we worked really, really hard."
The visitors took a two-goal half-time lead, which was a touch harsh on Sunderland, who had earlier hit the bar through Sebastian Larsson before Giaccherini pulled a goal back in the 52nd minute.
Try as they might, the Black Cats could not find an equaliser and were undone by a counter-attack as Suarez clinched victory for Liverpool in the closing stages.
"We spoke at half-time and I thought you saw a great reaction. They gave it a really good go in the second half," said Sunderland's interim head coach Kevin Ball, who was taking charge of his first Premier League match following Paolo Di Canio's sacking last weekend.
"We took it right to the death but unfortunately you're susceptible to that counter-attack where 3-1 - and I don't mean this as disrespect to Liverpool - flatters them a little bit.
"It was a shame because I thought the lads had a real good go today and that's something we need to take heart from."