Iran had to twice come from behind to earn a draw in their opening World Cup match, an entertaining Group G encounter that featured a brace from New Zealand's Elijah Just.
"That may be our best performance ever, or since I've been involved with the All Whites, which is a long time," Bazeley said.
"Overall, I thought it was a really strong performance and I think we showed the world who we are and who our players are."
The draw means all four teams in the group have one point after Belgium drew 1-1 with Egypt earlier on Monday.
It also means New Zealand - in their third appearance at the finals - remain without a win at a World Cup after seven matches.
Bazeley said his team produced enough quality moments to snatch a victory but acknowledged it hurt to come so close to that elusive win.
"We came here to win, and today we were so close to making history," he said. "We haven't won a game at a World Cup, and we're so close to doing that today, and that hurts."
New Zealand looked like they would hold on until Iran's Mohammad Mohebbi headed in a perfect cross to level, a moment Bazeley attributed to one of his team's few defensive lapses.
Nevertheless, he predicted his players would create more positive moments during the tournament.
"We've been building and building over the few years in regards to performances," he said.
"And I've said a lot every game we play, we have moments within the games against good teams where we play really well and create chances. Today we were more consistent with that."
FIFA World Cup 2026
The 2026 World Cup is taking place from June 11th to July 19th in the United States, Canada and Mexico. The tournament features 48 national teams and is played at 16 modern stadiums.
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