Palace failed to deal with Bertrand’s cross and Jeremy Pied played the ball back into the box for Long to sweep the home from 12 yards.
It was his first goal for club or country since February.
Hodgson’s side have endured a gruelling festive schedule and back-to-back home games against Arsenal and Manchester City had clearly taken their toll.
After the break, McCarthy had to claw out a fierce strike from Benteke after the big Belgian had turned Maya Yoshida.
And the forward had a hand in Palace’s leveller 21 minutes from time, flicking on Townsend’s chipped pass for McArthur to rifle home.
Sapped of any confidence, Pellegrino’s side were there for the taking and Milivojevic’s classy finish from outside the box gave the visitors a lead they never looked like losing.