Sergio Ramos headed in a 93rd-minute equaliser for Real, who went on to claim their record 10th European crown with a 4-1 win after extra-time against Atletico Madrid in Lisbon on Saturday.
Gareth Bale, Marcelo and Cristiano Ronaldo struck in the additional period against a tiring Atletico side.
Ancelotti said he always believed his team would get back into the encounter, labelling the win a 'dream come true'.
"We thought we could tie until the last moment," the Italian said.
"We fought long and hard and it was easier after the goal.
"It is a dream come true. The happiness is very great. I dedicate this title to all my family."
Ramos said his goal was his most important, but dedicated it to the fans and his family as Real won their first European title since 2002.
"This is my most important goal ever. The satisfaction we've been able to give our fans is incredible, we've fought right up to the end," he said.
"It's not just my goal... it belongs to all Madridistas, to my family and everyone who has supported us up to the end. We really deserve this after so many years waiting.
"Explaining how I felt is inexplicable. It happened so fast, it didn’t sink in. We've made history at a massive level. We've brought the Champions League to everyone that loves football."
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