The Netherlands had 32 shots - compared to their opponent's eight - but they were unable to beat Stephan Anderson in the Danish goal and were consigned to defeat by Michael Krohn-Dehli's first-half strike against the run of play.
"It was the only dangerous action of Denmark," said Van Bommel.
"I'm speechless, because these three points are very important.
"Five of us had chances. Once we hit the post. So many good opportunities."
The Netherlands' next fixture is against Germany in Kharkiv on Wednesday.