Two goals early in the second half from Dani Osvaldo and Rickie Lambert secured the three points for Saints, who moved in to the top four of the Barclays Premier League with the win.
After a first half of very few opportunities it was Osvaldo who grabbed his first goal for the club moments after the restart with a Lambert free-kick two minutes later proving enough to kill off Palace.
Pochettino spoke before the season about challenging for Champions League qualification in the next couple of years and he believes that the start to the season can be maintained.
"We are not setting ourselves any limits," he said.
"We just have to keep on believing and the players believe fully in how we want them to play. We are now going to have to work harder because teams know more about us and will respect us a little more now."
With Osvaldo and Lambert netting their first goals since being partnered in attack and his defence again proving impenetrable Pochettino hailed the perfect day for his side.
"I think it is complete happiness for every manager is what happened today, a clean sheet and your two strikers getting on the scoresheet," he said.
"In the second half we were much better than the first half and over the 90 minutes we were the better team.
"We struggled a bit in the first half to understand the set-up of the game but after the break we corrected certain things and that allowed us to play much better in the second half."
The Morocco international was free on goal but was booked by referee Martin Atkinson when he went to ground as he looked to round Saints goalkeeper Artur Boruc.
It was unclear whether Chamakh was clipped with Holloway not convinced his frontman took the right decision.
"When I was stood on the line I thought the referee had got it wrong," the Palace manager said.
"Obviously watching it now he didn't. He [Chamakh] went down far too easily for my liking. I can't understand why he didn't chip it or shoot.
"He felt he could take it past the goalie and he felt there was some contact on him but as an away team you are never going to get that are you. He had a great chance to score but he probably didn't make the right choice."
The result leaves Palace with three points from their first six games since returning to the top flight.
"I didn't like the way we capitulated again really," Holloway said.
"We had just let a goal in and it hadn't been a problem until then. We looked much more like a unit and were winning it back much easier than we were last week.
"The lads stuck to what they were doing and unfortunately on a couple of counter-attacks we had chances to maybe even get in front.
"This division is so harsh and so hard. Getting used to it and believing you can play in it can be difficult. We haven't improved quick enough or early enough and we look like we need a goal."
- Sports & Recreation
- Rickie Lambert
- Barclays Premier League
- Marouane Chamakh