Palmer says Chelsea players love and want to fight for Pochettino
His comments come amid speculation about Pochettino's future at Stamford Bridge. The former Tottenham Hotspur and Paris Saint-Germain coach has one more year under contract at Chelsea.
"For it to be in our hands to get sixth place is a big thing for us," Palmer told the BBC. "Chelsea are a big club and everyone wants to finish higher than sixth but you have to look at where we were at the start of the season.
"All credit to the manager. All the players love the manager and want to fight for him."
Chelsea secured all three points on the road with a 2-1 victory over Brighton & Hove Albion, thanks to Palmer's opener, which was his 22nd league goal in a spectacular first season with the team.
Since being appointed in May last year, Pochettino has taken Chelsea to the final of the League Cup, which they lost 1-0 to Liverpool, as well as the semis of the FA Cup, where they were beaten 1-0 by Pep Guardiola's City.
Despite a long list of injuries at Chelsea - new signings like forward Christopher Nkunku, defensive midfielder Romeo Lavia and captain Reece James have missed much of the campaign through injury - Pochettino said was proud of his team for overcoming setbacks.
"So happy for the players, they deserve the full credit for the way that we worked, our approach to the game and the way we tried to apply all of the work we were doing on the training ground," the Argentine told reporters.
"I am proud. To face a team like Brighton is always difficult here. We are so happy because the three points mean we are closer to achieving what we want."