Braga ultimately proved they had too much class in front of goal to leave Chris Hughton's side with an uphill task if they are to qualify for the knockout phase.
Two goals from Helder Barbosa, early in the game and again late on, sunk the Blues after Marlon King had offered up hope following a controversial effort from Lima that gave the visitors a 2-0 lead.
It was a much-changed City side that took to the St Andrew's pitch, with Hughton making six following Saturday's 3-0 Championship win over Millwall.
This was always going to be a tough opener for the Carling Cup winners, although they should have been up and running inside four minutes.
Charging down the right wing, King laid off the perfect pass for Adam Rooney, another of Hughton's summer signings, the 23-year-old plucked from Inverness Caley Thistle.
Remarkably, when a sweeping effort with his right foot appeared on the cards, Rooney instead snatched at it with his left, side-footing a gilt-edged chance into the side-netting.
There was a high price to pay for the miss because within 90 seconds Braga had opened the scoring in stunning fashion.
From the left, Helderson flicked in a curling cross to the edge of the area where Barbosa thundered a 20-yard left-footed volley beyond Boaz Myhill.
If there was nothing Myhill could do about that effort, the Blues goalkeeper conjured a superb save shortly after to effectively keep his side in the game.
Myhill was at full stretch in tipping a 25-yard free-kick around his right-hand post from former £8.5million Newcastle star Hugo Viana.
At that stage Braga threatened to walk away with the game, but Blues remained true to their task, and instead produced two chances of their own to suggest they should have been level at the break.
The first was relatively comfortable enough for Quim to deal with, for although Nathan Redmond's right-footed shot had pace and curl, it was at a safe height for the veteran goalkeeper to fist clear.
Then in the 21st minute Rooney was inches away again, flicking over the crossbar a left-wing free-kick from Wade Elliott.
For the remainder of the half and for almost the first 15 minutes of the second, the game had little to commend it as it was scrappy and with any moves continually breaking down in the final third.
That was until the 59th minute when Braga grabbed their dubious second.
From the left wing, Elderson delivered a piercing cross into the area for Nuno Gomes, with Lima at that stage a good yard offside, although not interfering with play.
The ball, however, fell fortuitously into his path of Lima, and he needed no second invitation to lash a 10-yard half-volley into the roof of the net.
Hughton made an immediate double substitution, bringing on Burke and Wood, the latter playing a pivotal role in giving City hope.
Minutes after Myhill had made another oustanding fingertip save to deny Lima, Burke produced an angled run and low pull back into the six-yard box for King to tap home into an empty net.
Redmond forced a stunning save from Quim and David Murphy lofted a 20-yard free-kick over the bar before Braga hit City on the counter three minutes from time to seal the points, with Barbosa firing a low pass from substitute Carlos Silva into the roof of the net.