Wales, whose goalkeeper Wayne Hennessey (35) was sent off late in the game, had looked set to grab a point after fending off a spirited Iran side but stoppage-time goals by Roozbeh Cheshmi (29) and Ramin Rezaeian (32) handed them a huge blow.
The Welsh sit bottom of Group B on one point, while Iran are second on three points.
England, who are top with three points, and the United States, on one, are due to meet later on Friday.
Wales, who drew 1-1 with the US in their opener, face a do-or-die clash against England on Tuesday as they look to stay in the World Cup, where they are playing after a 64-year wait.
"I am disappointed because we fell below the standards we set, that got us success and qualification for the World Cup," Page told reporters.
"If we'd have met those standards and made those performances we would have been alright. But we fell below that and when you do that in top competitions you get punished.
"Iran posed a constant threat. We just could not get going and that really disappoints me the most."
Page said he was not happy with the team's performance even before the sending off of Hennessey, who became the first player at this year's World Cup to pick up a red card.
"We want to finish the competition on a high, if it is in our hands to go through. We want to finish with a good performance and a win," Page added of Tuesday's England game.
"We are low at the minute but we are going to get back up tomorrow and get ready for another tough game."
Captain Gareth Bale (33), who became Wales' most-capped player with 110 appearances, said the players were gutted.
"It is going to be difficult for us for sure. We will have to see. What can I say? We recover and we have to go again," said Bale, who had little impact in the Welsh attack.
