Bale won the late penalty that he converted in emphatic style as Wales, playing their first match at the finals for 64 years, drew 1-1 with the United States.
In doing so, Bale - who missed a spot-kick against Turkey at Euro 2020 - became the first Wales player to score at a World Cup since Terry Medwin in 1958.
But Page revealed that he had checked on Bale to see whether the Los Angeles FC forward could complete 90 minutes, with fitness concerns surrounding him heading into Qatar 2022.
"Gareth looks after himself," Page said. "Irrespective of what he does at club level, whenever he puts that jersey on for Wales he gives performances that surprise me.
"He certainly surprised me again. I said to him on about 75 minutes, 'Are you OK to carry on?' He went, 'Yes I'm fine'.
"At the end of the game he said, 'That's why I stayed on, for the penalty'.
"He manages himself, he manages himself through games. If he had gone full tilt from the first whistle he probably would have got to half-time."
Bale had played only 30 minutes of football for LAFC since completing Wales' Nations League game against Poland in September.
That was the first time he had played 90 minutes for over a year, and he said after LAFC's MLS Cup final victory earlier this month that he was dealing with a "few little issues" over his fitness.
Page said: "We have evolved Gareth's position. He can still play out wide, but I like to give him the freedom to come inside and play off the front man.
"He is very good at finding those spaces. He's intelligent, he's got a wise head.
"So he puts himself in those positions in the box to be able to get us penalties.
"Defenders commit to making tackles in the box and it was a definite penalty. He used all of his wisdom there."
Page paid tribute to Nottingham Forest defender Neco Williams, who said after the USA game on social media that his grandfather had died on Sunday.
Williams dedicated the point to his grandfather and said he had "got through it from the support of my team mates and family".
Page said: "Neco is a very talented footballer and I was very pleased with him.
"He plays like a winger and is quite aggressive. I only took him off at the end because he was really tired and has not played many games lately.
"We changed shape to match the USA and get more attacking players on the pitch.
"As a whole I was pleased with our defensive shape, albeit we were deep in the first half.
"I wanted us to be more aggressive and press higher up the pitch.
"Neco's a big part of what we do. He gives us energy and he's very good from an attacking point of view."
Iran, beaten 6-2 by England in their group opener, are Wales' next opponents on Friday.