The US deservedly led at halftime through Timothy Weah's calm finish which looked set to give them victory until Bale was fouled by Walker Zimmerman and netted the 82nd-minute spot kick.
Bale helped Los Angeles FC win the MLS Cup this season but has yet to play a full match for the Major League Soccer side and entered the World Cup having played only 28 minutes of competitive football since September.
"Irrespective of what he does at club level, he manages to put the jersey on for Wales and give performances that surprise people. And he certainly surprised me today," Page told reporters.
"I said to him on about 75 minutes, 'Are you OK to carry on?' And he went: 'Yes, I'm fine.' And then at the end of the game he said, 'That's why I said I'm fine.'
"He manages himself through games, so if he had gone full tilt from the first whistle, he probably would have got to halftime. So he's very experienced and clever in managing his body."
Bale's goal was his 41st at international level and Wales' first at the World Cup for 64 years as they picked up their first point at the finals since their last appearance in 1958.
A switch in tactics by Wales at the break to introduce target man Kieffer Moore in place of Daniel James changed the game and allowed them back into the contest, with their pressure paying dividends and culminating in Bale's penalty.
"First half was not how we wanted to play and the United States played really well," Bale said. "We played quite poorly but (there was a) tactical change at halftime and, yeah, (we) came out very strong in the second half."
"The goal was exactly what we needed. It didn't matter who scored it, but we needed it to get back in the game. It was important to do that, and it gave us a foothold then to try and go and win ... We were unlucky not to win the game.
"It felt like we played very well in the second half. We were on top most of the game. Personally, it was great to get a goal, but like I said, the most important thing is the team scores."
