"That said, we were still right in there when we got the equaliser, it was still wide open.
"We had a big chance at the end as well to equalise. As always in these games, it comes down to fine margins, but I think they have been the best team in the tournament and I think over the piece, they deserved to win.”
Southgate brought Palmer on with 20 minutes remaining, with the midfielder having an instant impact as his first-time strike levelled proceedings just three minutes later.
Prior to that, England skipper Harry Kane had been replaced by Ollie Watkins.
The manager added: "Physically, it’s been a tough period for Harry. He came in short of games and not quite got up to the level we would have all hoped.
“The games are hugely demanding and we felt Ollie’s freshness could help us press. I thought the two substitutes that came on did what we wanted them to really."