Playing at the Ernst Happel stadium where Spain beat Germany 1-0 to win Euro 2008, Atletico took the lead in the eighth minute when Koke sent Felipe Luis clear on the left and he squared for Raul Garcia to tap into an empty net.
Brazil-born Costa showed why Spain coach Vicente del Bosque wants to call him up when he made it 2-0 in the 20th after racing from inside the Atletico half, dancing past the last defender and slipping the ball into the corner of the net.
The home side tried to force their way back into the match but Costa dashed any hopes of a comeback when he controlled an Emiliano Insua centre and struck from close range eight minutes into the second half.
Atletico have a maximum nine points from three matches, with Russia's Zenit St Petersburg on four after their 1-0 win at Porto. Vienna have one point in fourth and last place, two behind the Portuguese side.
Costa was making his Champions League debut having been suspended for Atletico's opening group wins against Zenit and Porto after picking up a red card in last term's Europa League.
The 25-year-old, who holds dual Brazilian and Spanish nationality, is the fourth Atletico player to net a double on his first Champions League appearance after current coach Diego Simeone, Sergio Aguero and Juan Esnaider.
"We played a great game," Costa, who is the La Liga top scorer this term with 10 goals in nine games, said on UEFA's website.
"I felt good out on the pitch and was very happy to make my Champions League debut in such a manner," he added.
"Our way of thinking is to take things on a game by game basis, trying to reach our maximum level in each encounter."
Under Argentine Simeone, who has transformed the club since taking over at the end of 2011, Atletico have made a blistering start to their domestic and European campaigns.
The 2010 and 2012 Europa League winners won their opening eight La Liga matches before suffering a first defeat of the season when they went down 1-0 at Espanyol on Saturday to slip one point behind champions Barcelona.
Tuesday's success marked the first time Atletico have won three consecutive matches in Europe's elite club competition since their debut in the 1958-59 season.
"It was a very good first half for us in terms of how we interpreted the way in which we had to control the game and its rhythm," Simeone said.
"We cannot afford to move away from this game-by-game mentality," he added. "We know the difficulties we will face in the Champions League and La Liga."
Vienna's wait for a first Champions League goal continues, although they came close to scoring when Philipp Hosiner volleyed against the underside of the bar in the 37th minute.
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