Sergio Ramos' controversial finish put Zinedine Zidane's side ahead before Atletico striker Antoine Griezmann missed a penalty.
However, substitute Yannick Carrasco sent the Champions League final into extra-time with a close-range finish.
Cristiano Ronaldo struck the winning penalty after Juanfran had missed.
How the Champions League final was won: Relive the drama from the San Siro
How the penalty shootout unfolded
In the shootout, Lucas Vazquez, Marcelo and Gareth Bale scored for Real, with Griezmann, Gabi and Saul Niguez replying to make it 3-3.
After Ramos stepped up to make it 4-3, Juanfran hit the post, leaving Ronaldo with the chance to win the cup.
The competition's all-time leading scorer kept his composure to win it - his third individual Champions League triumph following on his from his successes with Manchester United in 2008 and Real in 2014.
Zidane's tactics deliver silverware
Zidane, a Champions League winner with Real as a player in 2002, replaced Rafael Benitez as manager at the Bernabeu in January, with Real still reeling from a bruising 4-0 hammering by Barcelona earlier in the season and facing the prospect of a second successive campaign without major silverware.
The 43-year-old finishes it as the seventh different manager to deliver a European Cup to Real's hierarchy. As stylish as his side were going forward, they were just as impressive at the back as Atletico's attacking duo of Griezmann and Fernando Torres were starved of service before the interval. Bale and Ronaldo repeatedly dropped back into midfield to help cut out the supply routes to Griezmann and Torres, while Karim Benzema also typified Real's work rate by dropping back to defend.
Atletico were the better side after the break yet fortune was on Real's side as Pepe's rash challenge on Torres went unpunished and Stefan Savic stabbed another golden chance wide.
What now for Simeone and Atletico?
Three weeks ago, Atletico were on course for a La Liga and Champions League double, yet Diego Simeone's side end the season without a trophy after the most heartbreaking of defeats.
Having finished three points behind champions Barcelona in Spain, they are left to reflect on what might have been despite an impressive second-half performance at the San Siro.
Griezmann became more dangerous as the final wore on but his missed penalty proved crucial, even though Carrasco later levelled after Juanfran's brilliant volleyed cross.
It is the third time Atletico have reached the final and failed to win yet the future remains bright, with Simeone likely to remain in charge next season despite interest from Premier League clubs.
His task this summer will be keeping his assets, with Griezmann and midfielder Saul having been linked with moves to Chelsea and Manchester United respectively.
Clattenburg emerges with praise
England's Mark Clattenburg had his work cut out as he refereed a tempestuous derby, which saw eight bookings.
Real defender Pepe was at his theatrical best from the start as he tried to influence the Premier League official, who also had to calm down Simeone after Dani Carvajal's early foul on Griezmann.
Ramos put Real ahead from what looked like an offside position but Clattenburg's assistant kept his flag down.
However the 41-year-old from County Durham, who will be officiating at Euro 2016, was right to award the penalty which saw Griezmann hit the underside of the bar in the 47th minute.
Congratulations to real Madrid for the un decimal.
