The Dragons were aiming to keep their top-two destiny within their own hands at the Ahmed bin Ali Stadium, but they produced a woeful showing against Team Melli and succumbed to two injury-time goals.
Having had goalkeeper Wayne Hennessey sent off for a high tackle on Mehdi Taremi in the 86th minute, the 10 men of Wales held out until the eighth minute of second-half injury time before Rouzbeh Cheshmi drilled a terrific long-range effort into the bottom corner.
Any slim hopes of Wales rescuing a point were then dashed with 100 minutes on the clock, as the Dragons committed bodies forward and were hit on the break, with Ramin Rezaeian lifting the ball over Danny Ward for Iran's second.
Defeat for Wales leaves them at the bottom of Group B after two games, meaning that Page's side must secure victory over England next Tuesday if they are to have any hope of progressing to the last 16.
Speaking to BBC Radio Wales after the game, Page insisted that the Dragons' efforts against Iran before Hennessey's red card were "not acceptable", and he took responsibility for the defeat.
"Our performance before the red card was not acceptable. We hoped it would all come together today, but we were nowhere near the levels we've set and the standards we've set in recent games," Page said.
"One or two you can carry, but when there are so many off days there is only one outcome. I always take responsibility, that's my team, the sending off didn't help of course but at that point we had five forwards on the pitch so couldn't get a defensive shape.
"But yes I'll always take responsibility for performances. It's a difficult changing room at the minute but we will be ready [against England], it's a local derby and a great game to finish on and we'll be prepared to go."
Iran attacker Ali Gholizadeh had a goal ruled out for offside in the 15th minute of the match as Team Melli won the midfield battle with ease, but they had to be extremely patient for the opening goal.
Gholizadeh and Sardar Azmoun hit the post within a few seconds of each other in the second half before two late goals sparked pandemonium, and Carlos Queiroz was left delighted after his side stopped the "bleeding" from their opening 6-2 defeat to England.
"Two points. We played today with amazing character, which is the profile of our team. We play with a sense of unity, cohesion. After the first game we need to get back on route," Queiroz told reporters at his post-game press conference.
"The second point was to get rid of this bad feeling of bleeding. Football is a game of different moments. It's not because you win or lose.
"Sometimes you lose your dignity, your honour, but of course in our first game we were bleeding in our pride. This was an opportunity to rebound. We put a good scarf [bandage] to stop the bleeding. We played brilliantly and deserved to win."
Iran have now gone level on points with England before the Three Lions meet the USA this evening, and Queiroz's side will hope to advance to the knockout rounds of the World Cup for the first time in their history when they face the Stars and Stripes next Tuesday.