The win lifted Liverpool back into the top four with 42 points, five behind leaders Manchester City.
Yet Rodgers acknowledged that if they are to remain in the Champions League places they need to improve on a defensive effort that allowed Stoke to come back from two goals down to draw level at 2-2 before halftime.
Lax defending had allowed former Anfield players Peter Crouch and Charlie Adam to cancel out a Ryan Shawcross own goal and Suarez’s first.
“I was a bit frustrated,” Rodgers told the club website (www.liverpoolfc.com).
“I said at halftime that for 38 minutes it was the perfect away performance; solid, controlled, we played good football and looked like we could score goals.
“But we lost our concentration and we didn’t work hard enough in that period to manage the game through to halftime and then all of a sudden, when you play against good players, they can score goals, and that’s what happened.
“I didn’t want to focus too much on that at halftime, to be honest. I said to the players that if we keep making mistakes like that we will lose games. But I think we’ve shown in my time here that we’re a real offensive threat and we can get goals. I thought then the response in the second half was brilliant.”
Steven Gerrard and Suarez restored Liverpool’s two-goal advantage but that was halved when Jonathan Walters was the beneficiary of a weak attempted save by keeper Simon Mignolet.
Sturridge put any hopes of another Stoke comeback to rest with a fine finish from a Suarez cross, and in the process raised hopes that the pair’s relationship would once again flourish.
“Both of them combined really well,” Rodgers said.
“They’re both outstanding players and outstanding talents. You could see from their link-up for Luis’ goal, where Dan has slid it round the corner to him and Luis has finished it really well.
“Then there’s Dan’s movement for Luis to play him in, which was exceptional.”
(Reporting By Josh Reich; editing by Toby Davis)