The only goal of game arrived on 18 minutes, when Liverpool defender Sotirios Kyrgiakos brought down Mossoro to concede a penalty.
Midfielder Alan stepped up to take the spot-kick, and confidently beat Pepe Reina to his right to put the home side ahead.
It was a lead Braga very much deserved. They dominated the opening period against a strangely lacklustre Liverpool, who struggled to put together any possession and all too often looked to an isolated Dirk Kuyt up front.
Meanwhile, Braga - through in particular Hugo Viana and Rodrigo Lima - continued to pose problems to the Liverpool defence and threatened to increase their lead.
On 38 minutes defender Silvio reacted first to a Liverpool clearance and crashed a superb volley against the bar. It was a warning sign for Liverpool, and fortunately they took heed after the break.
Dalglish introduced his £35 million signing Andy Carroll early in the second half, and the striker immediately injected some life into the Liverpool effort.
Joe Cole had a penalty appeal waved away just after the hour mark, and Kuyt's superbly struck volley drew a sharp save from Artur in the Braga goal soon after.
Meanwhile, Braga's threat gradually eased away. Having already beaten Arsenal and Celtic at home in the Champions League this season, the Portuguese side looked to have settled for 1-0.
And so it stayed, ending Liverpool's unbeaten record in the competition this season, and sending the teams to Anfield next week with the tie intriguingly balanced.