Atletico's top scorer got the visitors off to a fine start with a well-taken finish after five minutes and his tireless work rate proved a nuisance throughout, as Diego Simeone's men moved back above Valencia and to within six points of pacesetters Barcelona.
Griezmann set the tone for a convincing display, smashing home his 15th league goal of the season after pouncing on an error in midfield.
Although Cordoba were afforded plenty of the ball, Atletico were in control in their first match since Simeone signed a contract extension last month.
Saul extended their lead before the break with a neat header and, while Cordoba improved in the second half, the away side rarely looked troubled.
Bebe did almost pull one back for the hosts when his long-range effort struck the crossbar in the 77th minute, but Atletico ultimately cruised to all three points, inflicting a 10th successive defeat on Cordoba and re-opening a four-point gap ahead of in-form Sevilla.
Saul urged his Atletico team-mates not to take struggling Cordoba lightly in the build-up and the champions began full of purpose.
It took them just five minutes to break the deadlock, as Griezmann seized on Bruno Zuculini's tame pass, ran at the nervous Jose Crespo and rifled into the bottom-right corner from 20 yards.
Despite taking an early lead, Atletico were by no means entirely dominant, allowing Cordoba plenty of time on the ball.
The visitors were the only side who looked like scoring, though, and they next went close in the 23rd minute, Jose Maria Gimenez heading just over from Koke's corner.
Atletico did eventually double their advantage six minutes before the break, as Saul arrived late in the penalty area and met Gimenez's flick-on with a brilliant header that gave Cordoba goalkeeper Juan Carlos no chance.
Jose Antonio Romero introduced on-loan Valencia attacker Fede Cartabia at the interval and the Argentine made a difference, as the hosts became more fluid in the final third.
Romero's half-time tweaking also brought more from Nabil Ghilas who, after twisting and turning in the penalty area, was denied a goal by a crucial Gimenez block in the 54th minute.
Simeone opted to afford Griezmann a rest for the final 20 minutes after running himself into the ground and that change seemingly encouraged Cordoba.
The hosts went desperately close to clawing one back with 13 minutes left, as Bebe's powerful drive from 35 yard smashed against the crossbar.
Cordoba enjoyed some late pressure to excite the home support, as Jan Oblak saved well from Crespo, but it ultimately proved to be too little too late.
