They will play either Scotland or Ukraine in June and are now one game away from reaching the World Cup finals in Qatar.
Wales - extended their unbeaten run at home to 17 games - opened proceedings with a sensational free-kick by Bale.
The Welshman scored another incredible strike, this time with his left foot and despite conceding an own goal in the second half, they held on to win 2-1.
Wales boss Page told Sky Sports: "I don't think it's sunk in probably yet. What a game, what a night, what an occasion, what an event.
"The star man turns up and I've just said to him, 'That's the best free-kick I've ever seen in my life'. Two great goals. A deflected goal against makes it an anxious last half an hour but to a man I thought they were outstanding.
"We had to dig deep and dig a trench at the end. They thoroughly deserve all the plaudits they're going to get those players. They're an unbelievable group of men."
On June's play-off against Scotland or Ukraine, Page said: "We're a cup final away now and that's it."
Bale added on Sky Sports: "I had some cramp at the end but I'll run into the ground for this country."
The forward was branded a parasite in Spanish paper Marca this week as he continues to remain on the books in Los Blancos despite not playing.
Bale - who is out of contract - labelled the paper 'disgusting'.
"I don't need to send a message. It's a waste of my time. It's disgusting and they should also be ashamed of themselves. I'm not fussed. End of."
Ramsey praised his team-mate, adding: "It's not been easy for him but him coming and playing for his country means everything and you could see that with the way he plays.
"We're a good group here, we all get on very well and we want to work hard for each other. Then he can produce moments of magic like that so we're grateful to have him here."