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.
Cristiano Ronaldo scores the winning penalty in the 2016 Champions League final
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.
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 doing his defensive duties.
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.
Bale shines ahead of Euro 2016
Bale was excellent as Real dominated the first half - powerful going forward and quick to help out at the back when required.
Gareth Bale has won the Champions League on two occasions since moving to Real Madrid in 2013
The Wales international was a danger at set pieces while he was heavily involved as his side took the lead.
Despite having an effort cleared off the line, Bale faded after the break and he spent most of extra-time limping around the pitch with cramp, unsurprising considering his work rate.
Yet he still put himself forward for the decisive shootout - and scored with a cool, low finish to help his team to a famous win.
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.
Yannick Carrasco celebrates scoring for Atletico Madrid against Real Madrid
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 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 earlier reported 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.
Gareth Bale was a contender for his marauding runs while Yannick Carrasco helped swing the match back in Atletico's favour after the interval. However, Casemiro played a crucial role for Real, the Brazilian making eight successful challenges during the final
The stats you need to know
Atletico have now contested more European Cup/Champions League finals without winning the trophy than any other club (three).
This was the eighth Champions League final to go to extra-time, and the seventh to go to a penalty shootout.
Zinedine Zidane is the second man (after Miguel Munoz) to win the European Cup/Champions League with Real Madrid as both a player and a manager.
Zidane is the first French manager to win the Champions League.
Sergio Ramos became the fifth player to score in two different Champions League finals, and the first defender to do so (also Raul, Samuel Eto'o, Lionel Messi and Cristiano Ronaldo).
Indeed, Ramos joins Messi and Eto'o as one of three players to score in his first two Champions League finals.
Antoine Griezmann became the first player since Arjen Robben in 2012 to miss a penalty in the Champions League final (excluding shootouts).
Only Marcello Lippi (three) has lost more Champions League finals as a manager than Diego Simeone (two).
feeling fabulous at Stadio San Siro.
feeling amazing at Stadio San Siro.
