Wales finished second in their qualifying group and secured home advantage in the March semi-finals by holding the world's top-ranked side to a 1-1 draw in Cardiff on Tuesday, with United defender Rhys Norrington-Davies an unused substitute.
By doing so, Wales have avoided meeting fellow top seeds Italy, Portugal, Russia, Scotland and Sweden in the last four, although they could come up against one of those nations in a play-off final to decide qualification for Qatar 2022.
Wales will now host a semi-final against either Austria, Czech Republic, North Macedonia, Poland, Turkey or Ukraine as Europe fills its final three qualifying spots.
For the play-off finals, also played over one leg, a draw is conducted in advance to determine which semi-final winner will play at home.
Former Barnsley and Rotherham United striker equalised for Wales, who were without captain Gareth Bale through injury.
And Page, who played over 100 times for the Blades and took charge of the Welsh national team in Ryan Giggs’ absence, said: "When our fans are rocking like they were in the last two games we'll go up against anybody. We'll respect the teams but we don't fear anybody. Bring it on.
"I did try to play home advantage down to take the pressure off the players.
"But they're not daft, they're intelligent enough to understand the importance of it."
Wales have lost only one competitive home game since being beaten by the Republic of Ireland in 2018 World Cup qualification four years ago.
Denmark claimed a Nations League victory at Cardiff in November 2018, but Wales have since won 11 and drawn five of their 16 home matches.
Page, who said he was a "wreck" in the post-match huddle when addressing the squad, added: "Our supporters have been exceptional in the last two games.
"When they were singing the anthem at the end, their support for us, I've been there as a player myself and it absolutely helps
"We look forward to the home tie now. They were fantastic and certainly helped us achieve that."
Wales' only appearance at a World Cup came in Sweden in 1958, when John Charles and company reached the quarter-finals.
On that occasion Wales reached the finals via a two-legged play-off victory over Israel and Page added: "The players know what this means to us as a country and as an association.
"To be part of a World Cup would be unbelievable. We tried to take the emotion out of it, play the game and not the occasion. Inevitably we were going to get caught up with it, but they dealt with the pressure.
"That group of players in there, the camaraderie they've got, the team spirit, it's unbelievable. It's like nothing I've ever experienced before. To be part of that makes me truly proud to lead them."
Page also represented United’s weekend opponents Coventry City in his playing career, before hanging up his boots and turning to coaching.