Last season's runners-up fell to their fourth top-flight loss of the campaign as substitute Ayoze Perez netted a second-half winner for the in-form hosts.
Alberto Moreno failed to clear a cross from Moussa Sissoko and Perez, who also netted the decisive goal in Newcastle's 2-1 triumph at Tottenham, was on hand to fire home from close range.
"It's a game we had almost full control of," Rodgers told BT Sport. "Maybe we didn't create as many opportunities as we would like, but we get punished buy a mistake and it's cost us the game.
"I think we passed the ball well, the options in front of the ball were good, but we probably lacked that little bit of quality in the final third."
Mario Balotelli was handed a start after coming off the bench to score the equaliser against Swansea City in the League Cup on Tuesday, but the Italian emerged late for the second half - barely arriving before the whistle blew.
Rodgers, however, played down the incident.
"It's not about Mario, it's about the team," he added. "If I talk about Mario I've got to talk about every player.
"We've collectively lost as a team, we've got to analyse it and move on. It [Balotelli's late arrival in the second half] had no bearing on the result."
The loss leaves Liverpool well off the pace in the title race and Rodgers pleaded for patience from the fans, insisting things would improve.
"Progress this season was always going to be a slow process because of so many changes," he explained. "We haven't started how we would like, but we will improve.
"Understandably for supporters it will be a difficult time. We can only continue to work hard. It [patience] is always the key, especially when results aren't going as you would wish."
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