Gronkjaer, who Blues are hoping will agree to a move to St Andrews having had an undisclosed bid for the winger accepted by Chelsea, replaced the injured Dennis Rommedahl on 24 minutes with the scores level at 0-0.
The Danes took a first half lead through Jon Dahl Tomasson and two minutes from time Gronkjaer marked his first game of the tournament with a goal after playing a neat one-two with Tomasson.
Gronkjaer had missed Denmarks first game of the tournament against Italy because of the death of his mother.
The Danes now face Sweden, who drew 1-1 with Italy, in their final group game on Tuesday.