The Turkish midfielder scored the second in the 64th minute and set up substitute Leo Baptistao for the third in the 80th, after Zenit had fought back strongly to level at 1-1 early in the second half with a thunderous shot from Hulk.
Atletico, who won the Europa League in 2010 and 2012, made it 13 wins from 14 in UEFA competitions at the Calderon overcoming a side who came into the clash unbeaten in 10 games in all competitions.
"We dominated the first half in terms of terrain but without many chances on goal, controlling their counter-attacks, but it was a different game in the second half," Atletico coach Diego Simeone told Canal Plus.
"They got the equaliser and could have had a second when they hit the crossbar, but the team flicked a switch and got the second goal and the third finished the game off."
Atletico moved to the top of the group, level on points with Porto who won 1-0 at Austria Vienna, but Simeone shrugged off any talk of them being favourites to finish in first place.
"Porto and Zenit have more experience than us in this competition. For many of our players this is their first experience of it, so we are only going to think about the next game," he said.
Zenit started very deep with a five-man defence looking to hit Atletico on the break, and the first half was played almost entirely on the edge of their area.
Atletico's Brazilian defender Miranda broke the deadlock with a powerful header at the near post from Koke's corner in the 40th but the Russians reacted well.
Zenit coach Luciano Spalletti withdrew a defender in place of a midfielder, and the switch paid off as Thibaut Courtois brilliantly palmed away Aleksandr Kerzhakov's header.
Courtois could do little about the equaliser, however, as Brazilian striker Hulk wound himself up and battered a long-range shot into the top corner in the 58th, and soon after Kerzhakov hit the crossbar with a free kick.
Atletico restored their advantage when Arda pounced to bundle in a loose ball from another Koke free kick in the 64th.
The extravagantly-bearded Arda then fed Baptistao as he sped off the flank, and the Brazilian-born Italian stroked the ball into the far corner.
"The game was just as Simeone said it would be, very intense," the 21-year-old Baptistao told Canal Plus, after making his debut in Europe's elite club competition.
"It was my first game in the Champions League, hearing the hymn and wearing the shirt with the symbol, it was very emotional. It wasn't just another game."
Courtois made another sharp save at the end but Spalletti was disappointed with his side's performance overall.
"We played badly in the first half," the Italian said. "Later we were better and took the initiative in the game but Atletico scored the second and ended the game.
"Atletico are a great team and their players have won all the individual battles."