The stubborn Portuguese side, who have never won their domestic title, earned a famous goalless draw at Anfield to progress to the quarter-finals 1-0 on aggregate.
Liverpool manager Kenny Dalglish could not hide his frustration after his team struggled to break down the visitors on a night of few chances and paid the price.
"Over the two legs it was a penalty kick that decided it (in the first leg). That's about right. There was not much between either side," Dalglish told a post-match news conference.
"They were very well organised and we knew before the game that the responsibility was on us to score a goal."
Their exit means Liverpool's only route back into Europe is a fifth-place finish in the Premier League but they are six points behind Tottenham Hotspur, who also have a game in hand.
Liverpool striker Andy Carroll, making his first start since a 35 million pounds move from Newcastle United in January due to injury, failed to open his account.
FEW CHANCES
"Maybe if Andy Carroll's header goes in during the first half it would have been a different outcome," said Dalglish.
"Overall we are pleased to see him back on the pitch. We weren't disappointed with his contribution in any way. He was a bit unfortunate he didn't score a goal."
Liverpool created few chances and squandered those they did manage to carve out of a rugged Braga rearguard.
"You don't need to work the goalkeeper if you put it past him. But we didn't put anything past him," lamented Dalglish.
"You need to be strong creatively if you are going to break down a really organised side. Or you need a bit of luck.
"We certainly didn't have much fortune over the two legs but I suppose everyone would say the same thing.
"Like every other team in the world we could use a bit more creativity but we are no different to anyone else."
Asked about Liverpool's chances of playing European football next season, Dalglish said: "It was an option for us to get back into Europe next year if we'd won it (the Europa League).
"We know that in every league game we will have to try and get as much as we possibly can. We were going to do that anyway, so I don't know if it will be a blessing or not."
"You only get in Europe if you win games. If you win enough games and get enough points we will get in Europe."
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