An inexperienced Atletico side took the lead in the first half when Luciano Vietto slotted home a powerful header before Arsenal rallied after the break with a brilliant equaliser from impressive teenage midfielder Emile Smith Rowe.
Paris St Germain are the other team in the tournament and the French champions will face Arsenal at the same venue on Saturday before closing the event against Atletico on Monday.
Arsenal opted for a fairly strong line-up that included new signing Bernd Leno in goal ahead of Petr Cech, while Hector Bellerin, Aaron Ramsey, Alexandre Lacazette and Pierre-Emerick Aubameyang all started for the Gunners.
Atletico, meanwhile, were missing a raft of players involved in the latter stages of the World Cup and it was no surprise to see the Premier League side dominate possession and territory as they looked to utilise their pace down both flanks.
New coach Unai Emery also gave recent signing Matteo Guendouzi a central midfield role and the French teenager made a confident start with his compatriot Lacazette drawing a pair of smart saves from Jan Oblak as Arsenal pressed for an opener.
The La Liga runners-up were content to soak up pressure and look to pounce on the counter and took the lead in the 41st minute.
Angel Correa was the architect of the goal when he burst down the right-hand side and cut back a driven cross that Vietto met with a powerful low header that flew past Leno before the German keeper could react.
Cech was introduced at the break as both sides changed their goalkeepers and Arsenal equalised almost immediately when 17-year-old Smith Rowe broke forward and fired home from long range past Atletico replacement Adan.
Arsenal's goalkeeper was soon called into action when Guendouzi lazily lost possession on the edge of the box and Cech made a superb double-save from Kevin Gameiro and Rodri as Atletico looked to regain their lead around the hour mark.
The match became stretched as both sides searched for a winner following a host of second-half substitutions but neither were able to find the decisive strike until Atletico prevailed 3-1 in the shootout.
Adan saved brilliantly from Henrik Mkhitarayan, Joe Willock and Eddie Nketiah before he stepped up to rifle home the winning penalty off the crossbar.
"I thought we done really well under a new manager. We pressed well and created plenty of opportunities but just couldn't put them away," Arsenal captain Ramsey said in a post-match interview.
(Reporting by John O'Brien; Editing by Keith Weir)
