However, Pep Guardiola helped him not only settle but realise his potential. The Bayern boss told Costa he could become a difference-maker in Munich.
"He also told me about [Lionel] Messi," the Brazilian revealed. "He said 'not to become Messi, because there's only one Messi.'
"'But you can do other things, things nobody else can do.'"
Guardiola wasn't wrong. Imbued with the Catalan's confidence, Costa took the Bundesliga by storm, racking up 10 assists (and one goal), in his first seven appearances.
Unsurprisingly, he was unable to sustain such a remarkable ratio but by the end of his first season in Germany, Costa had established himself as one of the most feared wingers in world football.
Guardiola was in no doubt that a star had been born. "He'll be a top, top player for years to come," the former Barcelona boss enthused.
However, everything changed when Guardiola departed for Manchester City. Costa was optimistic that he would continue to develop under new coach Carlo Ancelotti but the pair never clicked.
So dramatic was Costa's dip in form under the Italian, that Bayern agreed to loan him to Juventus last summer for €6 million, with a view to a permanent transfer at the end of the 2017-18 campaign for an additional €40m.
Club president Uli Hoeness was happy to see the back of the Brazilian.
"Costa didn't work out because he was quite a mercenary whose character we didn't like", he told Frankenpost last August.
However, Bayern CEO Karl-Heinz Rummenigge was unsure about letting him go right until the end.
"I always found him an interesting player," he admitted. "All he would have to do is reach the same level again that he was playing at during his first season with us."
In the end, though, Costa was allowed to leave. It, initially, looked like a shrewd decision, as the 27-year-old initially struggled to adapt to Serie A.
However, Juve boss Massimiliano Allegri kept the faith and has slowly helped Costa get back to his brilliant best.
Indeed, going into Sunday's top-of-the-table clash with Napoli, Costa is the one Juventus attacker in excellent form.
Paulo Dybala was so poor in the midweek draw at Crotone that he might not even start in Turin, while fellow Argentine Gonzalo Higuain has now gone six games without a goal in all competitions.
Of course, both men have the requisite talent to get back among the goals against Napoli but if they do, Costa will probably be their creator.
He has set up Juve's last four goals, coming off the bench to rack up a hat-trick of assists in last weekend's 3-0 win over Sampdoria before teeing up compatriot Alex Sandro for the opener at Crotone on Wednesday.
All of this, remember, has come after he absolutely terrorised his international team-mate Marcelo for 90 minutes of Juventus' 3-1 victory over Real Madrid in the second leg of their absorbing Champions League quarter-final second leg at the Santiago Bernabeu.
That night in the Spanish capital, he was the Douglas Costa of Bayern, burning one player after another with his blistering pace before peppering the penalty area with dangerous crosses (one of which led to Blaise Matuidi making it 3-0).
Of course, Napoli have their own attacking threats. No player has created more chances in Serie A this season than Lorenzo Insigne (90), who was instrumental in the Partenopei's stirring come-from-behind victory over Udinese, a result which saw Maurizio Sarri's side draw to within four points of Juve at the summit of the standings.
However, there is no overlooking the fact that Napoli been flagging in recent weeks, a product of the fact that Sarri has rarely rotated his key players, combined with the strain of trying to sate the Neapolitan public's thirst for a first Scudetto since 1990.
Dries Mertens, in particular, has looked increasingly jaded of late, as underlined by his dismal return of just two goals from his last 10 outings.
Costa, by contrast, is fresh and firing on all cylinders. He has started just 15 games this term – Mertens and Insigne have started 31 apiece – and Juve are now reaping the benefits.
There is arguably no more effective player in the league right now.
Only Luis Alberto (12) has more assists than Costa (10), while only Alejandro Gomez (94) has completed more dribbles (91). Those are remarkable figures for a player that has amassed just 1498 minutes of game time.
What it underlines, though, is what Guardiola used to tell Costa on a daily basis. He can be one of the game's great difference-makers – just for Juventus now, rather than Bayern.