Kieran Trippier and Ivan Perisic scored in regular time, before the Juventus forward lost his marker in the 109th minute to net the winner and his second goal of the tournament.
The win puts Croatia into the decider against France on Sunday and Mandzukic, 32, said his country showed great courage in the come-from-behind win.
"This is a miracle. Only great teams can be as brave as we were and fight back from a goal down against teams like England. We've played our hearts out throughout the tournament," he said.
"We were like lions out there tonight and we will be the same in the final. We enjoyed the game."
Croatia boss Zlatko Dalic, meanwhile, did not believe it was any type of miracle for his side to top England, telling a press conference that his side were "in all aspects of the game", he was proud to make a bit of history for the nation.
"For Croatia as a country this is history," he said.
"I don't know whether a smaller country reaches the World Cup final. We are going to play England in the UEFA Nations League in September but we do not have a proper stadium to play that game.
"This is the kind of infrastructure we do not have. But we have our hearts, our pride and that is what is important to us.
"Those experts that thought that England would progress to the final are not experts! If they were experts, they'd know Croatia are a better team, and we have shown it."