European Under-21 champions England made a winning start to the defence of their title with a routine qualifying victory in Luxembourg.
Lee Carsley's Young Lions beat Spain 1-0 in July's final and won comfortably in their first 2025 qualifier.
Palmer added a third from 15 yards with the strike going in off the crossbar.
The win takes England top of Group F, with Azerbaijan, Northern Ireland, Serbia and Ukraine also in the same section.
Only the group winners are guaranteed a place in the 2025 tournament in Slovakia.
Three of the nine group runners-up will also advance automatically, while the other six nations that come second will enter the play-offs to decide the final three places.
England are next in action on 12 October when they entertain Serbia at Nottingham Forest's City Ground, before an away match four days later against Ukraine, with that game taking place in Kosice, Slovakia.