Portugal goalkeeper Diogo Costa had his heart in his mouth after two and a half minutes when Mika Biereth charged down his clearance, but he was able to scamper back and shovel the ball out before the debutant could turn it home.
Costa was then alert to a Jesper Lindstrom effort from range, shortly before Pedro Neto tried his luck from distance at the other end where Kasper Schmeichel made a fine stop.
Midway through the half, a blocked Biereth shot fell kindly for Christian Eriksen, who caught the ball sweetly on the volley, only for his strike to be blocked by the hand of Renato Veiga, leaving referee Irfan Peljto no choice but to point to the penalty spot.
Eriksen stepped up, but Costa guessed correctly and made a great save down to his right. Approaching half-time, Biereth and Eriksen were left wondering what they had to do to score - the striker was denied again by an inspired Costa, before the Manchester United man beat the goalkeeper but not his club teammate, Diogo Dalot, who cleared off the line.
After the break, Costa continued to single-handedly keep Portugal’s hopes of a second Nations League trophy alive by denying Gustav Isaksen with a fingertip save and clutching an Eriksen free-kick.
Having survived that spell, Roberto Martínez’s men began to dominate the possession and territory but were failing to test Schmeichel.
That came back to haunt Os Navegadores 12 minutes from time, as just moments after missing the target, Rasmus Hojlund was teed up by a fine Andreas Skov Olsen pass, allowing him to slot the ball past Costa for his first of the campaign.
Portugal rarely threatened an equaliser and had no shots on target in the second half until the 95th minute, as Cristiano Ronaldo remained ineffective for 90 minutes.
No wins in their last eight on foreign soil for Denmark will, however, give Portugal hope of turning the tie around in Sunday’s second leg in Lisbon.
Flashscore Man of the Match: Rasmus Hojlund (Denmark)