Some children sat on the shoulders of their parents to get a better look as the bus went by.
The giant white bus was emblazoned with the phrase "Campeones 13" and the club's crest to mark their record-extending 13th European Cup success.
Team captain Sergio Ramos and other players waved and filmed the crowd with mobile phones from the top of the bus, which was escorted by police on horseback.
Ramos took the walkway set up over the plaza's fountain and draped the statue of the goddess Cybele with the club's scarf and flag as the crowd cheered.
He and Real Madrid left-back Marcelo then held up the trophy over their heads as white confetti rained down and the Queen anthem "We are the Champions" blared out from loudspeakers.
"I had to be here and celebrate this win with them," Ivan Gonzalez, a 23-year-old student who wore a pink Cristiano Ronaldo jersey and a Real Madrid scarf wrapped around his neck, told AFP at the plaza.
'Pride and ambition'
The bus then made its way to Real's Santiago Bernabeu stadium where a celebration with fireworks will later be held.
Roughly 80,000 people packed the stadium on Saturday night to watch Real's 3-1 Champions League final win over Liverpool on eight giant screens.
The crowd erupted in delight at the final whistle, breaking out in familiar chants of "Campeones, campeones".
Earlier on Sunday at a reception at the headquarters of the Madrid regional government, Ramos joked that winning the Champions League was "turning into a routine".
"In the difficult moments, when people doubted us, this team showed it has pride and ambition, that it does not get tired of winning," he added.
"I think this is the key of this success, along with having a great coach." Real's "decimotercera," or 13th title in the competition, moved them six clear of second-placed AC Milan in the all-time honours list, after lifting the trophy for a remarkable fourth time in five seasons.
Former French international Zidane also became the first coach to win three consecutive Champions League crowns, despite having spent less than three years at the helm.
The squad also stopped by Madrid city hall where they presented mayor Manuela Carmena with a jersey with the number 13 on the back that was signed by the players.
Carmena, who attended the match in Kiev, said Real had won thanks to the "magisterial leadership of its manager and the synergy of all its players".
Spanish Prime Minister Mariano Rajoy, a die-hard Real fan, sent a telegram to the club's president, Florentino Perez, to congratulate the team for their victory, which he said had "revalidated" the club's "continental leadership".
"Thank you for this sporting show that filled all sports lovers with emotion and pride, and for once again taking Spain's name to the highest levels of world sport," he added in the telegram, his office said in a
Rajoy had planned to watch the match in person in Kiev but he cancelled the trip at the last minute after the main opposition Socialist party filed a no-confidence motion against his government in parliament.
Real's victory made the front page of all main Spanish newspapers on Sunday.
"Madrid dominates Europe with a legendary team," headlined top-selling daily El Pais below a photo of Ramos holding up the Champions League trophy surrounded by the rest of his teammates.
"An eternal Real Madrid," wrote rival daily El Mundo while conservative daily ABC headlined: "A Real Madrid of legend".