"We showed more experience in international competition. We were very organised and took our goals with merit," Shakhtar coach Mircae Lucescu said.
Braga, who had already sampled the rigours of the Champions League when they were thumped 6-0 by a rampant Arsenal in their debut in the tournament's group stage this month, were this time outdone by Shakhtar's patient and efficient approach.
The Portuguese started strongly and almost made the perfect start when Moises's header was cleared off the line by Razvan Rat in the third minute.
Shakhtar enjoyed plenty of possession but looked slow and failed to threaten, allowing Braga to use their speed on the counter-attack.
Winger Alan put Leandro Salino through on the half-hour and the midfielder forced Pyatov to make a fine block with his foot.
Moments before half-time, striker Matheus led a counter-attack after a Shakhtar corner, racing down the right wing to set up Salino, who wasted another fine chance by placing his shot just wide.
The Ukranian champions, who lifted the UEFA Cup in 2009, took advantage of Braga's inexperience to catch them on the break themselves in the second half.
"I knew Braga was going to take more risks in the second half and give us more space to counter attack," Lucescu said.
Striker Luiz Adriano opened the scoring with the Ukrainian champions' first effort on target, firing in a diagonal shot from inside the box on 56 minutes after a clever pass from his compatriot Douglas Costa.
Braga pushed forward for an equaliser and almost scored five minutes later when Matheus's bicycle kick was palmed away by Andriy Pyatov, who reacted quickly again moments later to block Paulo Cesar's powerful drive.
"At 1-0, yes, the lack of experience weighed on my team. Suffering a goal that way, the team went looking for the equaliser in a disorganised, anxious way and you could see some inexperience," Braga coach Domingos Paciencia said.
Shaktar's Brazilian duo combined again smoothly in the 72nd minute to add their second, with Costa again feeding Adriano in the box. The winger then crowned a fine display by converting an injury-time penalty.
Braga, who knocked out Celtic and Sevilla in the qualifying rounds, face an uphill struggle in the group after Arsenal also kept their 100 percent record by beating Partizan Belgrade 3-1.
"This game was very important, maybe even decisive for us," Paciencia added.
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