The pressure, though, eased on Mauricio Pochettino after his Tottenham team came from behind to beat Hull 2-1 in the Premier League.
As Liverpool manager, Rodgers had pledged not to make the same mistakes as Tottenham when it attempted to rebuild after losing Gareth Bale in 2013. But, without Luis Suarez this season, the new-look Liverpool has failed to recapture the form that saw the team finish second in May.
Liverpool is now 12th after 12 matches — closer to the relegation zone than the top four.
Although Rickie Lambert finally scored his first goal for Liverpool, Dwight Gayle levelled from close range before Joe Ledley netted and Mile Jedinak scored directly from a free kick to secure Palace's first win since September.
"You've seen a team low on confidence today," Rodgers said at Selhurst Park. "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."
It seems unlikely Liverpool will be challenging for anything of note this season. The team's return to the Champions League after five years could end in the group stage unless its fortunes turn around on Wednesday against Ludogorets.
"Overall, that intensity and togetherness in our game isn't there," Rodgers said. "When you don't get the results, that affects you. That leads to (damaged) confidence. "
The loss leaves Liverpool 18 points behind leader Chelsea and just two in front of Palace, and Rodgers knows his own future could soon come under scrutiny.
"I'm not arrogant enough to think that I will be in a job through anything," he said.
Pochettino has also been feeling the heat in his debut season at Tottenham, which had lost three of its last four league games before travelling to Hull.
When former Tottenham player Jake Livermore put Hull ahead with an early 25-meter drive, it looked like being another miserable day for Pochettino.
But Tottenham had a man advantage from the 50th minute when Gaston Ramirez was sent off for needlessly wrestling Jan Vertonghen to the floor.
And striker Harry Kane then levelled with his 10th goal of the season in the 61st, meeting the rebound after Christian Eriksen's free kick came off the post. And the Denmark midfielder grabbed the winner in stoppage time from the edge of the penalty area after being setup by Aaron Lennon.
Hull is now just a point from the relegation zone, while Tottenham is up to 10th, only two points from fourth place.