Rodgers' side have been chasing the 23-year-old England international since that disastrous summer transfer window in 2012, which closed without the Reds securing any much-needed fire power.
Wednesday's deal was confirmed by the Anfield club, but the exact fee - believed to be in the region of £12m - was not disclosed.
Sturridge told Liverpool's official website that he was "humbled and happy" about the move.
"Brendan Rodgers said he sees me here for a long time and I also see myself here for a long time. I've not signed here to play for a couple of years and then move on," he said.
"I've signed to be here for as long as possible. It's a humongous club - for me, one of the biggest in the world - and to have the fans and world-class players we have here is amazing."
We can still achieve our objectives this season, so when you can do that going into the second part of the season it gives you real hope.
Rodgers, who turns 40 later in January, took up the reins at Anfield seven months ago.
Since then, he has faced a baptism of fire with some fans unconvinced of his credentials after just one previous season in the top flight with Swansea - the team he guided to the Premiership from the first division and successfully kept there in their debut season.
Liverpool supporters expressed further discontent after the largely unsuccessful summer transfer negotiations, although a lack of available funding at the time was said to be a contributing factor.
On Tuesday, as the January transfer window opened, Rodgers spoke out about the relationship with the club's board of directors.
"The support the club will give us will be excellent," he said.
"The owners have been great and I have no complaints in terms of where we are as a club at this time."
Rodgers also remains confident that he can continue to make improvements at the club and complete the goals he set at the beginning of the season.
"I love the expectancy and challenge of making this great club great again. I believe as we grow and work more together, we will improve," he said.