The Red Devils continued their bad habit of conceding early, Alan with two first-half strikes, but the Mexican forward inspired a recovery to maintain momentum in Group H
Alan scored in the second and 20th minutes to rock the Red Devils but the Mexican's double, either side of a Jonny Evans equaliser, rescued all three points.
Sir Alex Ferguson followed through on his pre-match talk of implementing a diamond, Wayne Rooney captaining the side from its forward point and Michael Carrick beating Scott Wootton to a start at centre-back.
The makeshift nature of the defence showed, however, when Alan profited from badly disorganised marking to rise and head home within just two minutes from Hugo Viana's cross.
The visitors had arrived fearless and they doubled their lead on 20 minutes, this time exploiting Carrick. Eder skinned the England man on the left touchline and darted inside to cut a pass back into the box, where Alan stabbed home for his second.
Braga's two-goal cushion lasted all of five minutes and it was thanks to a smart piece of refereeing that United got their riposte. Serbian official Milorad Mazic indicated an advantage when Robin van Persie was fouled on the left flank but Shinji Kagawa picked up the loose ball and dinked a cross for Chicharito, who was onside at the far post to head into the net.
Both teams pushed for more, Van Persie firing wide with his right foot at one end before Alan nearly exploited more uneasy defending by volleying just past the post. Kagawa's movement saw him break past the back line but, just as he rolled it to the unmarked Chicharito, the Japanese was harshly flagged for offside.
Needing to seize control of an open contest after half-time, United had to replace the injured Kagawa with Nani, flattening their diamond formation accordingly. Possession did indeed come more easily but penetration slightly less so as the Braga rearguard held their own.
The breakthrough came from a poorly cleared corner, however, which fell to Evans. The defender scuffed his first attempt but it bobbled back to him and a scruffy finish was enough to beat Beto and send the game level.
Buoyed, United pushed for more and, after setting Nani up for a saved stinger, the excellent Chicharito got further reward for a superb performance. The Mexican powerfully headed home from a pinpoint delivery by Tom Cleverley, another bright spark in the turnaround.
Braga had looked deflated ever since Evans' euqaliser and never really threatened to get back into proceedings, even when David de Gea's risky pass to Cleverley opened things up. Ruben Amorim's first touch deserted him, however, and the opportunity fell flat. In the end, Sir Alex's men were comfortable in their victory.