The record six-time Europa League winners are 15th in the league standings on five points, one above the relegation zone.
It's Sevilla's worst start to a league season since being promoted back to the top-flight in 2001.
Marcos Llorente scored Atletico's first in the 29th minute, firing a shot to the right of the goalkeeper and into the back of the net.
Atletico's win lifted them to fifth on 13 points, five behind city rivals and leaders Real Madrid.
It was a solid and dominant performance by Atletico who bounced back after two consecutive losses against Bayer Leverkusen in the Champions League and Real in the league.
Coach Diego Simeone was able to revert to his typical defensive style thanks to having defenders Jose Maria Gimenez and Stefan Savic back from injury.
Atletico took the lead through Llorente after Sevilla lost the ball from a throw-in close to their own area.
Koke played the ball the left to Llorente who was all alone inside the box to slot the ball in, ending his LaLiga scoring drought that lasted 47 matches. His last strike in the league was on May 1, 2021, in Atletico's 1-0 win at Elche.
"I'm happy but [mainly] for what the team showed as a collective effort. It was a complete game," Llorente told DAZN.
"Scoring and winning is good but I'm happier for way we played. It was a solid performance in every level and we have to keep going."
However, he was not able to stop Morata from scoring early in the second half, when the he took advantage of a clever recovery by Cunha to speed into the area and lob the ball over the goalkeeper.
There was also a record for long-serving Atletico midfielder and captain Koke who made his 554th appearance for the club, the most of any player. He beat the previous best of Atleti great Adelardo Rodríguez, whose record of 553 had stood for 46 years.