The All Blacks came out on top 20-16 in Buenos Aires as they began what will be an abbreviated Rugby Championship campaign on a winning note.
But after first-half tries from Ngani Laumape and Brodie Retallick, and 10 points from the boot of Beauden Barrett, New Zealand was kept scoreless in the second half.
It was clinging on at the end, and Savea agreed when asked if it felt like a "war" against the Pumas.
Savea said: "It was, mate. It was bloody tough out there.
"The Argentinian players came out firing and just kept coming at us the full 80 minutes. I’m just proud of the boys for sticking in there.
"We just talked about the next moment and dominating the next moment, so when we made mistakes we'd just connect and talk about the the next moment and how we could dominate that, and hopefully gain momentum."
New Zealand will regroup for next weekend's home game against South Africa, and look to work out how Argentina caused it so many difficulties.
"They bought a bit of line speed on our attack and they put our skills under pressure," Savea said.
"And at times their pressure worked. We'll learn from that hopefully."