Defending a 100 per cent sequence at the King Power Stadium, Nigel Pearson's side were second best against a slick Palace side under the stewardship of caretaker boss Lennie Lawrence following the departure of Dougie Freedman this week.
The visitors could have been ahead in the first minute but Yannick Bolasie spurned a chance from just yards out.
It was a rocky start for Pearson's Championship leaders but Julian Speroni - making his 250th league start in goal for the Eagles - had to be alert to save a Dave Nugent effort as the Foxes briefly threatened.
A header from Andy King cleared the bar as the hosts rallied, but Kaspar Schmeichel did well to hold on to a Wilfried Zaha cross as Palace - nine games unbeaten in the Championship - displayed all their attacking intent.
And Lawrence's men took the lead after 23 minutes. Schmeichel failed to deal with a Mile Jedinak corner, feeling he was impeded as he went to claim the ball, and Delaney rattled home from 12 yards out.
Zaha tried his luck from 30 yards out, but Palace did not have long to wait for their second as Ramage headed home from six yards out after 28 minutes, latching onto Bolasie's assist.
It had been a poor start from the hosts, but it got worse 13 minutes from the break when an injured Lloyd Dyer was replaced by Matty James.
Jamie Vardy would have hoped for a better connection with a header that sailed harmlessly over the bar from close range 10 minutes after the break, and Leicester were forced into a tactical change after 63 minutes when Danny Drinkwater was replaced by Ben Marshall.
Time was running out for the hosts and a speculative shot from all of 35 yards from James showed how well Palace had snuffed out the Foxes' threat.
With 14 minutes left Anthony Knockaert was replaced by Jeff Schlupp, and Leicester finally registered in added time when King headed home.