The German club, which was on the verge of bankruptcy in 2005, suffered a 2-0 defeat in their semi-final second leg to Real Madrid as Karim Benzema and Sergio Ramos struck late on but held on to win 4-3 on aggregate.
Dortmund will find out tonight (Wednesday) if they will play German rivals Bayern Munich in this month's final at Wembley when Jupp Heynckes' side take a 4-0 first leg advantage to the Nou Camp.
But McClaren says that for now the plaudits belong to Dortmund, who last won Europe's premier cup competition in 1997.
"It's a thoroughly deserved victory, but what a finish," McClaren told Sky Sports News. "Over the two legs Dortmund have been superior but for that last flurry from Real Madrid.
"But they hung on; they put every defender on and played eight at the back, one in midfield and one up and they survived everything.
"This is an absolute dream come true for Dortmund who last season went out very early. This season they have learnt from their experiences, they've ridden their luck and in the end they deserve to go through to the final. It's a fantastic achievement.
"Jurgen Klopp has done an unbelievable job. A lot more will be said about this trainer. He is such a larger-than-life character and he has transformed this Dortmund team from one that could have gone out of existence to one that is into the final of the Champions League. It really is an incredible story."
Following Real's defeat, manager Jose Mourinho gave the clearest indication yet that he wants to return to the Premier League.
The former Chelsea boss could yet finish the season with silverware when Real take on city rivals Atletico in the Copa Del Rey final.
Mourinho will make an announcement on his future in the summer but on Tuesday hinted that he is ready to leave Spain, saying that he's loved in England, especially by one club. McClaren believes 'the Special One' will leave Real having seen his side outclassed by a far superior team.
"I think this was the last throw of the dice for him," said McClaren. "They've had a disappointing season although they've still got the Cup final.
"But this was the trophy they wanted. They lost it in the first game when Dortmund were absolutely fantastic. They won the battle on the field and Klopp has won the battle off the field.
"Not many people do that against Mourinho but he's done it, especially at half-time in the first leg when he turned it around and his side got three goals to put themselves in a position to win this game.
"When you are a club like Real Madrid with all your history and tradition and you haven't won the Champions League for so many years it puts extra pressure on you but you saw in the second leg that this is a team of individuals.
"As the game went on they were starting to lose shape and relying on Ronaldo, who was never in the game, whereas Dortmund won this game because they are a team.
"They were superior in every department and are probably the fittest team in Europe - and on transition they are they are the quickest team. They are so dangerous.
"They've got momentum at the present moment and they've got luck on their side too. They could win this but you can't write off Bayern Munich."
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