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The Danes have reached the semifinals for the fourth time in their history and will face England at Wembley on Wednesday for a place in the final.
It has been an emotional time for the team after star player Christian Eriksen suffered a cardiac arrest in their opening group stage game. He has since been discharged from hospital after an operation to implant a heart starter device.
Patrik Schick got a goal back for the Czechs but they bow out of the tournament after an impressive showing.
Denmark opened the scoring after five minutes when Delaney headed in from Jens Stryger Larsen's corner.
They had a great chance to double their lead 10 minutes later when Mikkel Damsgaard's shot looped over the goalkeeper but it was easily cleared away by the Czechs.
Denmark made it 2-0 just before the break when Joakim Maehle delivered an outstanding cross with the outside of his boot which was converted in by Dolberg from close range.
The Czechs grabbed a goal back four minutes into the second-half after Schick produced a fine finish from Vladimir Coufal's cross.
Schick is now leading the race for the Golden Boot with Cristiano Ronaldo on five goals.
With 10 minutes remaining, Vaclik was called into action once again to deny Maehle scoring from close range.
Antonin Barak went close to grabbing a late equaliser with a long range volley but it went wide and Denmark held on for the victory.