He's surprisingly big for 6ft 2in, mouthy and a bully. He picked up 11 yellow cards in 24 league games last season, excluding a four-game ban in the Europa League, given for head-butting a Viktoria Plzen player.
He infuriates opposing defenders. They kick him. They spit on him. He just brushes it off. Last season, Atletico travelled down to Seville to play Real Betis in the Copa del Rey, Spain's cup competition.
Just before half-time, the ball was lumped 60 yards up field. After it bounced, Antonio Amaya, the Betis central defender, headed past his own keeper, who had come out to collect it, straight into Costa's path. Costa passed it in to an empty net.
In the tunnel after the full-time whistle, Costa couldn't help himself. "He was shouting and thanking me for the gift," said Amaya. "If my team-mates had not held me back, I would have killed him. That shows what kind of person he is: he has no heart and no shame."
But it was Costa's heart that endeared him to Diego Simeone, the Atletico manager who has overseen a remarkable transformation at the club after taking over when the team was mid-table in December 2011, picking up clatter of trophies along the way, including the Europa League in 2012.
Tonight, Atletico face Barcelona in the first leg of the Champions League quarter-final at the Camp Nou, while sitting on top of the league table in Spain, a point ahead of Barca - with a genuine chance of becoming the first team to break the Barcelona-Real Madrid stranglehold on the Primera Division since Valencia won the title 10 years ago.
Atletico fans despaired when AS Monaco swooped for Radamel Falcao last summer, another in a line of star strikers - including Sergio Aguero, Diego Forlan and Fernando Torres - to leave the club in the last several years for greater wages (and a shot at glory) elsewhere.
Costa, when presented with the opportunity to leave, declined, reportedly turning down the chance to treble his wages by joining Liverpool last summer.
Chelsea are the latest club to covet him. But he has found a kindred spirit in Simeone, a notorious hard man during his playing days with Atletico, Inter and Lazio. Simeone was impressed when he first took a look at him. "When I saw him in training, I wanted to die," he said.
"He was unstoppable. Diego Costa transmits a strength that has a contagious effect on the rest of the group. He gives everything, but he has to learn to control himself. People say he plays at the limit, they also said that about me."
Simeone kick-started a career that had never properly taken off. Costa joined Atletico in December 2006, but was farmed out every season on loan to Spanish backwaters, from Albacete to Valladolid. In 2011, a proposed move to Turkey was scuppered when he suffered a cruciate ligament injury in his knee.
Simeone's faith has provided healthy dividends. As second fiddle to Falcao last season, Costa bagged 10 goals in the league.
This season, he has scored 25 in 30 games, and, according to Opta, has a better shot-conversion rate than Lionel Messi or Cristiano Ronaldo - one in three, compared to one in four for Messi, and one in six for Ronaldo.
Costa gets all kinds of goals - headers, bicycle kicks, but his trademark is to barge past with pace, as he did on Saturday night away to Athletic Bilbao in the league. He scored the equaliser; Atletico won 2-1 against fourth-placed Athletic.
He was put through by Koke, who enjoys a rare telepathy with him, following an interception. Costa was still 45 yards from goal when he gathered the ball in the middle, but broke away from Athletic's two chasing defenders before sliding the ball low into the bottom left-hand corner of the net.
Costa can use his size and physicality to play with his back to goal as a target man, and he excels at drifting away from markers, by dropping deep or slipping out to the wings, only to re-emerge at the right moment - never there, always arriving; like all good strikers.
Costa could not be arriving in Barcelona at a worse time for the hosts, who are vulnerable when faced with burly centre forwards.
Karim Benzema took them for two first-half goals (and almost three if it wasn't for a goal-line clearance from Gerard Pique) in four minutes during the clasico nine days ago.
It is another measure of Costa's chutzpah that he has chosen to forsake the country of his birth, Brazil, in the year that it will host the finals of the World Cup.
The striker made his debut for Spain last month against Italy, having played as a substitute in two friendlies last year for Brazil.
Costa has a Spanish passport, a Spanish daughter, and speaks better Spanish than most Brazilian players in Spain. He moved when he was 18 years old; he is 25 now. "He was formed here," says Vincent Del Bosque, the Spain manager.
Marcos Senna (Spain) and Deco (Portugal) forsook Brazil, too, in recent times, but both were peripheral to the national team's plans. Costa, particularly given the injury problems suffered by Fred, would have been a shoo-in for the Selecao this summer, given his blistering form. Spain have benefited from the short-sighted decision to leave him out of the squad for the Confederations Cup last summer.
Costa will face Xavi, Andres Iniesta and several of his new Spain team-mates tonight. Atletico and Barca played out scoreless draws in their last two meetings; both contests turned into dog fights.
Just the way both Diegos like it. It could fall to Costa to find the space to score, by fair means or foul.