Martins wrong footed the New Zealand defence on the slippery surface at Waikato Stadium and stroked the ball with the outside of his right foot past a despairing Nikola Tzanev to end the hosts' hopes of causing the upset of the tournament.
New Zealand had never made it to the knockout phases of the tournament before but gave Helio Sousa's side a massive fright after substitute Stuart Holthusen equalised in the 65th minute.
Raphael Guzzo had given Portugal the lead in the first half as his side looked in control of the game before Holthusen's goal.
The goal spurred the crowd of almost 10,500 into raucous chants of "All Whites, All Whites" and gave the New Zealanders the wistful hopes of grabbing an unlikely victory.
"When we scored that goal the place went wild," New Zealand coach Darren Bazeley said in a pitch side interview. "When that happens you want another one."
Portugal will now meet Brazil in the quarter-finals in Hamilton on Sunday after the South Americans beat Uruguay 5-4 on penalties following a 0-0 stalemate after extra time.
The five-times winners of the competition had dominated the game as Uruguay sat back and defended but were only able to separate themselves from their neighbours when Rodrigo Amaral blasted Uruguay's second penalty over the crossbar.
Brazil were perfect and when Gabriel Jesus slotted Brazil's fifth successful spot kick to settle the match he was mobbed by his team mates.
Germany continued their clinical march into the last eight after Werder Bremen forward Levin Oeztunali scored the only goal in the 19th minute of their victory over Nigeria in Christchurch.
Oeztunali, the grandson of former Germany striker Uwe Seeler, slammed a powerful left-foot shot through the hands of Joshua Enaholo from the edge of the box to set up a quarter-final with Mali in Christchurch.
Dostonbek Khamdamov ensured Uzbekistan at least matched their performance from the 2013 tournament by grabbing both goals after the break in his side's 2-0 victory over Austria in Whangarei.
Ravshan Khaydraov's side, who only qualified for the round of 16 by beating Fiji 3-0 on the last day of the group stage, will meet Senegal in Wellington in the quarter-finals.
(Editing by Sudipto Ganguly)
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