The Danes were well worthy of at least a point in Frankfurt as England again struggled to live up to their tag as one of the pre-tournament favourites.
Harry Kane had capitalised on a Danish defensive error to fire England into an early lead, but Morten Hjulmand's stunning strike in off the post equalised before half-time.
Andreas Christensen spooned over when unmarked from a corner and Pierre-Emile Hojbjerg curled inches wide as Denmark finished strongly.
"I can't say we are disappointed but it's a shame. There was a result there we could have gotten," said Hjulmand.
England ended Denmark's dream run to the semi-finals of Euro 2020 three years ago.
Since then, Hjulmand's men have struggled to match raised expectations.
They crashed out at the group stages of the 2022 World Cup and were held 1-1 by Slovenia in their opening game in Germany.
Hjulmand urged his side not to drop their standards as they look to secure a place in the last 16 against Serbia on Tuesday.
"We believed we could win. We played well but the most important thing is that we play like we did today in the upcoming games," added Hjulmand.
"I know we have been looking for this kind of game for some time. It's taken us a while.
"Today we showed how we really play and express ourselves - with quality, with fire - this is how we want to play."
England remain on top of the group on four points, two clear of Denmark and Slovenia.
Serbia are bottom on one point after snatching a 1-1 draw against Slovenia earlier on Thursday thanks to Luka Jovic's stoppage time header.
