Liverpool's Mohamed Salah celebrates after scoring a goal against RB Leipzig during their Champions League match on Wednesday. AFP

Elsewhere, Liverpool eased past RB Leipzig into the Champions League quarter-finals with a 2-0 win in Budapest courtesy of two second-half goals to give the Reds a 4-0 aggregate win.

Meanwhile, "We put on a good showing, it was a great performance, we had a lot of chances in the first half but we just missed the finishing touch which was a shame," Griezmann told Movistar Plus after Barca had 20 attempts on goal but only scored courtesy of a Lionel Messi thunderbolt in the first half.

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Messi then had a penalty saved by Keylor Navas as Barcelona fell short of repeating their remarkable comeback against PSG in 2017, when they lost 4-0 in the first leg in Paris but won 6-1 in the return.

The five-time European champions had not been knocked out before the quarter-finals since 2006/07 when, with Messi aged 19, they lost to Liverpool in the last 16.

Liverpool's Mohamed Salah celebrates after scoring his side's opening goal in Budapest, Hungary. AP

Coach Ronald Koeman admitted that the outcome could have been different had Messi's penalty in first-half stoppage time, with the score at 1-1, gone in rather than been pushed onto the bar by Navas.

Barcelona can still win domestic silverware this season -- they play Athletic Bilbao in the Copa del Rey final next month and are second in La Liga after going 16 games unbeaten.

However, Atletico Madrid won earlier on Wednesday to go six points clear at the top of the Spanish table.

It is the first time since 2004/05 that neither Messi nor Cristiano Ronaldo -- eliminated with Juventus on Tuesday -- will feature in the quarter-finals of the Champions League.

Nevertheless, Koeman believes Messi, who is out of contract at the season's end, is encouraged by the way a young Barcelona team is improving.

Meanwhile, repeating their rapid-fire double act from the first leg, Mohamed Salah and Sadio Mane again struck within four minutes to settle the tie and keep alive Liverpool's hopes of rescuing their miserable domestic season with European glory.

After six successive defeats at Anfield and dropping to eighth place in the Premier League, Jurgen Klopp's team looked relieved to be away from home, with the second leg also played in Hungary due to Covid-19 travel restrictions.

But while the Reds showed attacking intent from the off, they squandered a string of chances throughout the first half.

Unmarked at a corner, Diogo Jota should have opened their account in the 18th minute but the Portuguese striker headed straight at Hungarian goalkeeper Peter Gulacsi who tipped over.

Midway through the period, an acrobatic scissor kick by Thiago Alcantara released Salah on the left but his left-footed effort was blocked by Gulacsi with Mane unable to convert the rebound.

Then on the half-hour, the onrushing Salah couldn't finish from close range after meeting a cross by Trent Alexander-Arnold who was sent clear on the right by a slide-rule pass by Mane.

Roused at last by the narrow escapes, Leipzig began to venture beyond Liverpool's high line and managed to carve out some chances of their own.

Kevin Kampl forced a save from Alisson Becker with a curler from distance, while a minute later Swedish playmaker Emil Forsberg, back from injury, clipped a snapshot just wide of the post.

Still, with Brazil's Fabinho back in the starting side and directing affairs in midfield, the better chances fell to Liverpool.

Jota again fired straight at Gulacsi just before the break and then into the side-netting on the whistle after a slip by French defender Dayot Upamecano.

Leipzig, flying domestically on the back of six straight Bundesliga wins but nervy in Budapest, urgently needed a goal to rattle Liverpool but continued to lack penetration in the second half.

In truth, Julian Nagelsmann's men, who dumped Tottenham out at this stage of the competition last year and Manchester United in the group stage this season, rarely looked like claiming an even bigger scalp by turning the tie around.

Their last opportunity arrived on 65 minutes but Alexander Sorloth's header from a cross by Hwang Hee-chan spun over Alisson's fingertips only to rebound off the bar.

Soon after, Salah and Mane ruthlessly took advantage of the let-off by securing Liverpool's berth in the last eight.

In the 70th minute, the Egyptian fired a shot past Gulacsi, then four minutes later Mane turned a cross by substitute Divock Origi past the hapless Hungarian.