Custodio wasted an early Braga chance when he smashed a shot wide from the edge of the penalty area but they soon settled into a defence-minded strategy.
Braga held out until 44 minutes when Fredy Guarin crossed for Falcao to plant a header in the top corner.
Mossoro almost equalised when through one-on-one but Helton saved well.
Porto's victory continues their superb season, in which they have won their league title and remain on course for a Treble - they face Guimaraes in the Portuguese Cup on Sunday.
And coach Andre Villas-Boas becomes the youngest man - at 33 - to win a Uefa club competition.
But Porto, whose dominance in their domestic league saw them go unbeaten and finish 38 points ahead of Braga, were far from their best in Dublin.
With two sides from the same nation competing for the trophy, it was always likely to be a cagey affair.
As so it proved as underdogs Braga opted to keep things tight and flooded the midfield and defence.
The ploy worked well for them for 44 minutes - and they even had a decent chance very early on for Custodio - but it made for a tedious game to watch.
However, Braga were undone by carelessness in possession and then a lapse in marking - Alberto Rodriguez gave the ball away and the defence failed to pick up Falcao, who finished with style.
Villas-Boas celebrates 'amazing start' at Porto
Until that point Falcao had cut an anonymous figure, while his strike partner Hulk had been on the end of a number of very strong challenges by Braga.
Only 38 seconds after the interval, Braga should have hauled themselves level when substitute Mossoro was clean through on goal but he hit his shot at the legs of keeper Helton.
Mossoro and fellow substitutes Kaka and Albert Meyong did their best to rouse Braga and their side did respond to exert a degree of pressure on Porto for the first time.
But the number of chances were outweighed by the number of bookings - eight in total - as the game continued to be feisty rather than fluent.
Meyong did get a sight on goal but scuffed his shot from the edge of the box and the ball dribbled out of play.
Porto were able to comfortably defend and even looked the more threatening when they countered - Hulk sent a shot over and Fernando Belluschi went closer with a drive from distance.
In the end, one moment of clinical finishing proved enough to give Porto the edge in a game that failed to live up to hopes and expectations.
Porto coach Andre Villas-Boas: "Without a doubt this is a great feat. We have one trophy left for a memorable season.
"This has a huge emotional impact for us. Round after round, we kept believing more. Porto becomes an even more special club, even more unique.
"It adds another trophy, which is important and builds its future."
Braga coach Domingos Paciencia: "I am very proud, I lived unique moments here. These players also did, it's a beautiful story.
BBC Sport 2011