Page said: "It's a cruel game, that's what I've said to the players, one kick away from qualifying. It hurts.
"I thought we were the better team, my message to the players at half-time was we are the better team, we will win this game.
"It's so disappointing but look how far we've come, there's something good happening with this group.
"We rallied around Daniel James, he had the bravery to take that penalty, I am so proud of that group."
Wales captain Ben Davies also said: "It's a difficult one to take, we put everything into the game but in the end it wasn't to be.
"I don't know what to say, it was a good performance but in the end the result is the most important thing and that went against us tonight.
"They're a good team, there wasn't much between us tonight but in the end penalties it's cruel.
"It's an emotional one right now, the character shown from the players showed everyone loves playing for Wales."