The France international had endured an underwhelming start to life in the capital following his big-money move from Real Sociedad, but was influential against Cordoba, netting his first La Liga goal for the club.
Champions Atletico looked comfortable throughout and it came as little surprise when they eventually took the lead in the 43rd minute, as Griezmann saw his shot deflect in.
Cordoba pulled one back early in the second half through Nabil Ghilas, but the vibrant Griezmann restored Atletico's advantage soon after with an unstoppable header.
Mario Mandzukic, another of Atletico's close-season investments, added the third and second-half substitute Raul Garcia headed in number four in the final 10 minutes.
Ghilas doubled his own tally right at the death, but his second goal proved to be nothing more than a late consolation, with the win moving Atletico on to 23 points - keeping them immersed in the five-way tussle at the top of the table.
Despite coming into the match without a victory this season, Cordoba began brightly and went close to taking an early lead as Iago Bouzon headed just wide of the top-left corner.
Atletico swiftly took charge, though, creating their first chance in the sixth minute, as Arda Turan dragged a shot just wide after pouncing on a poor headed clearance.
Diego Simeone's team went even closer to finding the breakthrough soon after, with Koke making the most of Cordoba's slack marking before striking the crossbar from 25 yards.
But although Atletico dominated possession, Cordoba looked threatening on the counter.
The visitors broke just after the half-hour mark and Ghilas found himself released into the area, but the Algerian fired a tame effort straight at Miguel Angel Moya from a tight angle.
However, the deadlock was finally broken two minutes before half-time when Simeone's men took the lead.
After showing fine composure to evade a tackle in Cordoba’s crowded area, Griezmann saw his shot take a huge deflection off Luso and fly past the helpless Juan Carlos.
Cordoba looked encouraged at the start of the second half and they found themselves level in the 53rd minute.
Ghilas met Fede Cartabia's free-kick delivery with a fine header looping header to restore parity, but Atletico were ahead again soon after.
Griezmann darted ahead of his man at the front post and nodded Juanfran's teasing cross beyond Juan Carlos in the 58th minute, capping a superb display with a finish to match.
Atletico's third arrived just four minutes later, as Mandzukic headed in a Koke free-kick and they added a fourth for good measure in the final stages.
Garcia, who replaced Arda, produced delightful header to knock in Koke's lofted pass with 10 minutes left and, although Ghilas took advantage of some uncharacteristically poor Atletico defending to grab his second late on, the hosts held on for an easy win.
