With Mario Balotelli sidelined with a groin injury, fellow summer signing Lambert was handed just his second Premier League start and did not take long to take advantage of his chance.
But that was as good as it got for Rodgers' side as Dwight Gayle – who scored twice in the memorable 3-3 draw here last season – levelled after man of the match Yannick Bolasie saw a driven effort come back off the post.
The second half started slowly but two goals in a three-minute period were enough to seal a well-deserved and much-needed win for the Eagles and Yannick supplied Joe Ledley to put the hosts ahead before skipper Mile Jedinak bent home a 30-yard free-kick to seal a 3-1 win.
"It was bitterly disappointing," he said of the performance.
"You've seen a team low on confidence today. Not quite together as a team. We need to find a solution very quickly because it was very disappointing.
"I put the team out there, the best team to win us the game. We made a good start. You could see our passing was a wee bit tentative and then we make mistakes. Mistakes you wouldn't expect to see at a team that's supposed to be challenging.
"Overall, that intensity and togetherness in our game isn't there. When you don't get the results, that affects you. That leads to (damaged) confidence. We have to work harder but we go away bitterly disappointed with the result. At this moment, we're just not good enough."
The draw here last season was ultimately damaging to Liverpool's title tilt as Palace struck three times in 11 minutes to snatch an unlikely result.
Luis Suarez was left in tears on the pitch following that game and since his departure the Reds have struggled for the type of form which turned them into challengers.
Rodgers invested the money received for Suarez into his squad and now wants to see more from them, insisting he knows results must improve if he is to avoid facing questions about his future.
"I'm not arrogant enough to think that I will be in a job through anything," he added.
"Any manager will tell you that you have to win games and you have to get results, especially after how we've been developing as a football club.
"But I have a great communication line with the owners. We've been honest enough with each other, but ultimately you have to get results. You have to perform.
"There's no doubt as a manager you have to get results. That will support the confidence to the owners and you take it from there.
"I will only ever do my best. The best has seen us develop well. Now I need to fight even harder and take the responsibility, because as manager full responsibility comes down to me. Any pressure comes on to me.
"We've brought in very good players. Some of them are very young and are not going to be ready for a few years, but they're in now. We have to do better in our performance level."
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