The defending champions had begun their campaign with a 26-7 win over Scotland on Friday but despite leading the Portuguese 17-7 at halftime and 24-19 with minutes remaining they could not close out the win.
Portugal, who had suffered 22 defeats in a row to the Kiwis, had a chance to snatch an improbable victory but failed to convert Nuno Guedes' last-gasp try.
New Zealand, winners of 12 of the 15 Sevens World Series titles and reigning World Cup sevens champions, did bounce back from the shock by beating Australia 14-5 to qualify for Sunday's knockout stages.
"We wanted to put a bit of respect back in the jersey after the Portugal performance," New Zealand captain DJ Forbes said.
New Zealand coach Gordon Tietjens said his side had been lucky against the Portuguese.
"We just fell off tackles. You are going to get punished playing like that in sevens against anyone. To be honest we shouldn't have even drawn that game, we should have lost," he said.
(Writing by Patrick Johnston in Singapore)
