The South Americans, ranked 10th in the world, are heavily favoured to advance and were expected to swamp the All Whites, but they had only two real chances to score a crucial away goal as New Zealand captain Winston Reid marshalled his side's defence.
New Zealand goalkeeper Stefan Marinovic produced a goal-line save after a mixup in defence and then tipped over a header from Aldo Corzo in the second half as the home side held their shape and Reid mopped up any attacks.
All Whites midfielder Ryan Thomas had arguably the best chance of the match when his 85th-minute shot flashed just centimetres wide of the right-hand post to ensure both sides remained deadlocked for the second leg in Lima next Wednesday.
Copa América
Diaz scores injury-time winner as Colombia clinch third at Copa America
10/07/2021 AT 08:30
"It is a really strong result for us, big result and we are pleased with where we are at," All Whites coach Anthony Hudson told reporters. "But our feet are right on the ground.
We haven't achieved anything yet. We still want to get to Russia. All we have to do is score one goal and that puts added pressure on them. We are very confident we can go there and score.
Burnley striker Chris Wood started on the New Zealand bench after coming into the camp with a hamstring injury but was subbed on with 16 minutes left as the All Whites looked for a goal.
The Peruvians, under immense pressure to qualify for a first World Cup since 1982, dominated early, with midfielder Christian Cueva and forward Jefferson Farfan causing problems for the hosts' defence. Despite their superior possession, they had only one genuine chance in the first half when Marinovic pawed the ball away centimetres from the line after a defensive mix-up with Reid.
The Vancouver Whitecaps goalkeeper was also called into action in the 63rd minute when he pushed a powerful Corzo header from a corner over the bar.
Wood forced the only real save by Pedro Gallese with his first touch before Thomas had his chance to send the crowd of 37,034 into delirium when a loose ball fell to him. His shot flashed just wide.
"We deserved a little bit more," Peru coach Ricardo Gaerca told reporters through an interpreter. "We had control of the game but couldn't score.
The All Whites are very tough. They have a very good organisation, but we will improve. I know we can play better.
Copa América
Jesus sent off for horror challenge as Brazil progress to Copa semis
03/07/2021 AT 09:36
International friendlies
New Zealand withdraw from England friendly due to Covid complications
14/10/2020 AT 08:52