The new signing has praised the size and ambition of his new side, while the club president described the Spain international's arrival as "an honour"
The ex-Porto forward left the Vicente Calderon for a reported €60 million [£51.9m] this summer, with the Rojiblancos turning to Spain international David Villa to fill the void left by Falcao's departure.
Villa, who Atleti cost €5.1m [£4.4m] from Barcelona, is excited to be joining the Liga giants and wants to help the club succeed on continental and domestic fronts this campaign.
"I'm very happy," the striker told reporters at a press conference on Monday. "Atletico Madrid have showed me the affection I needed.
"It's an honour for me to be at a club this big. I'm going to give my all for Atletico. I'm very motivated for the coming season. I want to make the fans of this club happy, it's a huge challenge for me.
"I'm here to help a team that is already a winning team. I was happy at Barca [but] last year I didn't play as much as I wanted. That chapter ends, and a new, exciting one begins at Atleti.
"I'll do my best to make sure the team doesn't feel Falcao's absence. Atleti finished third in La Liga, won the Copa del Rey, are in the Champions League.. Joining the club wasn't a difficult decision.
"I've left Barca to play regularly and be in the national team, but not because it's a World Cup year."
Atleti president Enrique Cerezo was delighted to have secured the attacker - who had been linked with a move to Arsenal and Tottenham in the previous six months - and is looking forward to seeing the impact Villa can have up front.
"It's an honour to welcome one of the most important international forwards of all time," Cerezo said. "Many great strikers have played for this club, and the arrival of David Villa sees us continue in this vein.
"David, I hope you continue to score many goals in the Rojiblancos shirt. I hope that we can enjoy many great footballing moments together."
Following the packed press conference, Villa was unveiled to around 20,000 supporters at the Vicente Calderon on Monday.