Robinson, whose team lost their first 2015 European Under-21 Championship qualifier 3-0 in Cyprus in May, were drawing 1-1 at half-time but could not keep out a much improved Denmark after the interval.
Southend defender Adam Thompson's headed goal had cancelled out Youssef Toutouh's opener, but that was as good as it got for the hosts as Nicolai Brock-Madsen's brace and Rasmus Jensen's strike all arrived in the space of seven minutes.
Northern Ireland were good value for the half-time scoreline, keeping the ball well in midfield and passing confidently but lacking the cutting edge to create clearcut chances.
Toutouh's 17th-minute opener came when he was left unmarked from a free-kick, but the lead only lasted until the 24th minute when Thompson rose at the far post to nod home a Josh Carson corner.
The away side changed formation at half-time, moving an extra body up front and the gamble paid off.
Brock-Madsen restored the lead after 57 minutes and made it 3-1 five minutes later when cut inside from the wing and fired past Connor Brennan.
Moments later Denmark put together a neat move, finished off when Jensen tapped home following a good save from Brennan.