Costa scored the opening goal and then saw his shot deflected in off defender Alan Hutton as home supporters celebrated just a third league victory of the season by chanting Mourinho’s name.
But, with the nerves around Stamford Bridge obvious, it had taken a dreadful double error from the visitors to allow Costa to ease the growing concern among home supporters by shooting his side into a 34th-minute lead.
Goalkeeper Brad Guzan had repeatedly looked uneasy when placed under pressure by the home side and his hurried kick was mis-hit in the direction of Joleon Lescott who failed to control the ball, gifting it to Willian.
The midfielder sprinted into the Villa area and waited for the goalkeeper to commit himself before squaring for Costa, recalled after serving a three-game suspension, to convert the ball into a gaping net.
The goal had been badly needed for Blues boss Mourinho because, until that point, Villa had created as many, if not more, chances as their hosts.
Just two minutes before the opening goal, Tim Sherwood’s team had carved out their best to date, Jordan Ayew being allowed to surge down to the left-hand by-line from where his cross found Rudy Gestede who poked an effort just wide, under pressure from John Terry.
It was the latest example of uncertain defending in the home defence, where Baba Rahman was making his first start, and looking unconvincing in the process.
– Swift improvement –
Hutton skipped away from the left-back far too easily after 11 minutes, advancing into the area and seeing a low, driven shot kept out by the outstretched boot of Asmir Begovic.
There was also a low shot from Jack Grealish and direct free-kick from Ayew, both from the edge of the Chelseaarea, which required Begovic to make routine saves.
Mourinho, keen to see a swift improvement to his team’s worst start to a season in 37 years, had left out Eden Hazard, replacing him with highly-regarded teenager Ruben Loftus-Cheek.
And while Chelsea‘s defensive frailties were obvious, they at least looked capable of scoring in the opening half, with Loftus-Cheek having an early shot deflected behind and Terry just clearing the post with a set-piece header later on.
After the restart, there was an early moment of promise for Grealish, as he cut in from the left and flashed a cross-shot through the six-yard area, but within eight minutes, Villa were a further goal behind and, apparently, out of contention.
The goal was created by Cesc Fabregas who floated a long pass into the area for Costa who controlled the ball on his chest and took a touch before seeing his shot beat the wrong-footed Guzan via a heavy deflection from Hutton.
The manner of that second goal certainly seemed to dampen struggling Villa’s self belief and Chelsea pressed on in an effort to record an even more emphatic margin of victory.
Pedro curled a left-foot shot just over from a threatening position, Fabregas’s cross was missed by Hutton and, fortunately for the defender, also by Costa who had not expected the ball to reach him.
Any hopes of a most unlikely Villa recovery were blocked out, literally, by Cesar Azpilicueta as he threw himself in the way of a scrambled Lescott shot at a 71st-minute corner and, thereafter, it was simply a question of whether Chelseawould score again — Fabregas coming closest with a fierce drive that flashed across the face of goal.