The Eagles were written off as relegation certainties after losing seven of their first eight matches and lying five points adrift of safety when previous manager Ian Holloway quit Selhurst Park at the end of October.
But fast forward three months and Palace sit 14th, five points clear of the drop zone - and just a point behind Swansea in 10th.
Crucially, they have kept five clean sheets in the 11 matches since Pulis took over and Tuesday's win over Hull, courtesy of Jason Puncheon's first-half strike, was their fifth by a 1-0 scoreline.
"We're a lot more solid than we used to be," said Speroni.
"It gives the team a lot of confidence when we keep clean sheets. We are defending as a team, trying not to leave any spaces at the back, and that means we are going into every game knowing that we can win.
"Two or three months ago, who would have thought we would be 14th?
"We are on a good run and we are in a good position but we have not done anything yet. That five-point gap can disappear in two games and there is a long way to go."
Pulis has been desperate to strengthen his squad during the January transfer window but Palace have yet to bring anyone in ahead of Friday's deadline, while the current players seem to be doing their best to persuade the manager he need not bother.
Skipper Mile Jedinak said: "If some business does get done before the end of the window, then fantastic. If not, then we have to move on and go full steam ahead between now and the end of the season.
"We have made it difficult for teams and if we can continue doing that until the end of the season it will be massive. We have done well at big places before this season and I am sure we will have a game plan at Arsenal on Sunday."
Hull, by contrast, paraded their new £14million strikeforce of Shane Long and Nikica Jelavic.
The duo barely got a sniff of a chance against Palace but manager Steve Bruce saw enough to suggest they can fire the goals to keep City clear of trouble.
"I thought they were both terrific," he said.
"Jelavic tired because he hasn't played a lot of football but the two of them will be terrific for us, I'm convinced of that."
Bruce is considering appealing against Allan McGregor's late red card after the goalkeeper reacted angrily to a challenge by Stuart O'Keefe.