Liverpool striker Sturridge scored twice and missed a penalty in a 4-0 thrashing of Everton at Anfield on Tuesday.
The England international was then brought off just after wasting another chance to complete his hat-trick and was clearly unimpressed with Rodgers' decision.
The former Chelsea man apologised after the game and insisted he was frustrated with himself following his penalty miss.
"I apologise for my reaction," he said on BT Sport.
"I was disappointed by missing the penalty and felt I'd let myself down as well as the fans.
"The hat-trick would have been the icing on the cake and it's unfortunate.
"I was disappointed with myself. It was nothing to do with the manager and I'd like to apologise for my reaction if anybody took any offence. It’s not about me, it’s about the boys and a great victory."
Rodgers was delighted with the impact made by his striker, but called on the 24-year-old to keep a lid on his emotions.
"In fairness to Daniel, he was frustrated after not scoring the penalty," Rodgers said.
"We are very much a team and if anyone is unhappy we say to just keep it to yourself and speak to me the next day, but there is no problem with Daniel. He was brilliant tonight."
The Liverpool boss was pleased by the manner in which his side rose to the occasion.
There has been little to choose between the Merseyside rivals so far this season as they battle to secure a top-four finish.
Roberto Martinez fancied Everton's chances of winning at Anfield for the first time since 1999 ahead of the game, but there was a gulf in class on Tuesday as Liverpool swept their neighbours aside.
Steven Gerrard opened the scoring with a header and Sturridge scored twice in as many first-half minutes to stun Everton.
Luis Suarez added a fourth goal early in the second half and Sturridge wasted a chance to complete his hat-trick when he put a penalty into The Kop after Tim Howard had upended Raheem Sterling.
Rodgers was delighted to reap the rewards of taking the game to Everton as his side enhanced their chances of securing a top-four finish.
The Liverpool manager said: "It was a gamble to go with two strikers but that's the reason we took the risk. We got four goals and could've had six or seven.
"It was a brilliant demonstration of how to play under pressure."
