The striker, brought back into the fold after being frozen out by coach Paulo Fonseca in January, netted an early goal at the Estadio Municipial de Braga as Roma took control of the tie.
Braga played the majority of the second half with 10 men after Ricardo Esgaio was sent off for a second bookable offence, before Roma doubled their lead late on through Borja Mayoral to take two away goals back to Italy.
Roma made a flying start to the match and Dzeko marked his return to the starting line-up with a goal inside just five minutes. The Bosnian stepped ahead of defender Raul Silva to steer home Leonardo Spinazzola’s cross, after a powerful drive down the left from the wing-back.
Whilst Braga saw more of the ball, it was Roma who enjoyed the better of the attacks early on. Amadou Diawara saw a low shot blocked and Dzeko was intercepted as he shaped to shoot in the box after breaking the offside trap.
At the other end, Fransergio dragged a shot well wide from range and Spinazzola cut out Nicolas Gaitan’s dangerous through ball for Esgaio, as the hosts struggled to create.
The Giallorossi ought to have doubled their advantage on the half hour mark, as Dzeko released Jordan Veretout in the box, who in turn cut back for Henrikh Mkhitaryan. The Armenian took too long getting the ball out of his feet however and could not apply the finish.
Roma had the ball in the net again as Pedro latched on to a Dzeko flick to round the keeper and slot home, though the former Chelsea winger was offside in the build up.
Braga pushed for an equaliser before the interval and Andraz Sporar headed wide from a teasing cross, whilst Spinazzola sent Gaitan’s low cutback behind for a corner, though they could not find a way through.
The Portuguese club started the second period brightly and saw appeals for a penalty waved away as Roger Ibanez clattered into Sporar in the box, with the Brazilian defender injured in the process.
However, Braga’s task was soon made all the more difficult as Esgaio was given a second yellow card for a cynical foul on Gonzalo Villar, catching the Roma man with his elbow as he looked to race past him.
Roma sought to make the most of their numerical advantage and looked to have netted again on the hour mark, only for Mkhitaryan to have strayed fractionally offside as he side-footed home Pedro’s clever through ball.
Dzeko had the chance to kill the tie midway through the second period as he slid in to meet Spinazzola’s low cross at the back post, but had stretched too far and couldn’t get enough power to test goalkeeper Matheus.
Just as Braga began to threaten an equaliser, the match was sewn up by substitute Mayoral late on. The Spaniard tapped home from Mkhitaryan’s low cross after a silky Roma move, with quick passing opening the Braga defence up and allowing the unmarked substitute to net his 10th of the season.
Roma’s victory sees them take a healthy advantage ahead of the second leg next week, as the Lupi host Braga at the Stadio Olimpico on Thursday.