The 30-year-old lifted the frustration inside the Juventus Stadium with a double to end a 14-game barren run in the competition - which stretched back five years to a clash with Porto while playing for Manchester United - as Juventus went top of Group A.
His first came in the 59th minute, as he cooly fired home after a perfect backheeled pass from Kwadwo Asamoah, while the second arrived in the last minute via a curling free-kick.
Massimiliano Allegri's side controlled possession in the opening half but a lack of quality in front of goal saw Malmo, group-stage debutants, hold them at the break.
Tevez finally proved the difference, though, and the scoreline could have been even bigger had Fernando Llorente's close-range effort not been disallowed for offside by the assistant referee.
It was a tough evening for the Swedes in Turin, and their plight may yet get even worse after a second-half injury to top scorer Markus Rosenberg.
While Arturo Vidal and Andrea Pirlo were both missing for the hosts through injury, Allegri was able to call on Giorgio Chiellini after injury and suspension, but the defender had little to do in the opening half.
Instead, all the action was at the other end as Juventus toyed with their opponents.
Paul Pogba, Llorente and Asamoah all failed to convert chances, with the latter appearing all over the pitch in an impressive display that only lacked a goal.
He had the clearest chance of the opening period in the 19th minute when he met Stephan Lichtsteiner’s cross, but his header sailed harmlessly over the crossbar.
Tevez's first effort on target was then easily saved, before his strike partner Llorente was denied by a deflection.
The Spaniard received a pass from Pogba inside the penalty area, held off two defenders with his back to goal before spinning to shoot, but his effort struck a defender and was diverted away for a corner.
With half-time approaching, the deadlock should have been broken as Lichtsteiner was released behind the Malmo defence. He squared a low cross to the back post but his team-mates failed to anticipate it and the danger was cleared.
The second half appeared to be going the same way as the first when Lichtsteiner saw two crosses well blocked, but a minute before the hour Tevez finally opened the scoring.
Having played a pass into the penalty area, the Argentine striker received a superb return ball from Asamoah before stroking it past goalkeeper Robin Olsen to end his Champions League drought.
Llorente thought he had doubled the lead a minute later when he tapped in a Lichtsteiner cross from close range, however the assistant referee's flag denied him.
Low crosses from Patrice Evra and Lichtsteiner saw Juventus go close to doubling their lead but when Pogba miscued a spectacular effort from outside the penalty area it seemed they would have to settle for a one-goal advantage.
With seven minutes remaining Olsen produced a heroic save to deny Llorente before Malmo, who had rarely been out of their own half, were caught on the counter-attack by substitute Alvaro Morata.
The youngster was felled by Filip Helander on the edge of the penalty area and from the resulting free-kick Tevez curled home his second in style to get Juve off to a winning start.
