The Merseysiders fell behind to a Raul Jimenez goal on the break in the 38th minute, the Mexican firing home confidently after Diogo Jota robbed James Milner.
Brighton kept a disciplined formation, frustrating Liverpool and as the second half wore on the home side began to lose their fluency.
Liverpool cranked up the pressure on West Ham United manager Slaven Bilic with a 4-1 victory at the London Stadium, while Burnley's remarkable start to the Premier League season continued thanks to a 1-0 victory at Southampton on Saturday.
Mohamed Salah had a remarkable first season with Liverpool, scoring 44 goals in all competitions and helping the club reach the Champions League final
Manager Juergen Klopp has extended his contract with Liverpool, less than a year since joining the club, the English Premier League side announced on Friday.
Barcelona were held to a frustrating 0-0 draw at Olympique Lyonnais in their Champions League last-16 first leg on Tuesday as both sides remained unbeaten in this season's competition.
Juergen Klopp's side, chasing their maiden title in the Premier League era and first in the top flight in 30 years, took a 14th-minute lead from a Virgil van Dijk header but had to wait until stoppage time to put the game to bed through Mohamed Salah.
IMAGES from the EPL matches played on Sunday.
Liverpool reached the Champions League semi-finals for the first time in 10 years as Mohamed Salah and Roberto Firmino struck decisive blows to secure a 2-1 win over Manchester City on Tuesday that completed a 5-1 aggregate victory.
Manchester City's Kevin de Bruyne sparked a late comeback 3-2 victory for his side at Newcastle United.
Liverpool were left frustrated by Championship side Blackburn Rovers who defended heroically to draw 0-0 at Anfield.
Liverpool forward Mohamed Salah on Wednesday called for an end to what he described as "massacres" in Gaza, saying humanitarian aid must immediately be allowed into the besieged Palestinian enclave.
Liverpool belatedly kick-started their year and damaged Tottenham Hotspur's slender Premier League title hopes as Sadio Mane struck both goals in a 2-0 victory at Anfield.
Liverpool will feel the weight of increased expectation to deliver silverware this season after making encouraging forays into the transfer market, manager Juergen Klopp said.
City are 16 points behind Liverpool - who have a game in hand - and Guardiola said he could not afford to be distracted
City are looking to retain the title they won in record-breaking fashion last year and secure their fourth Premier League crown in eight years.
Images from English Premier League matches played on Sunday.
European champions Liverpool are second in their Champions League group - a point behind Napoli - and Klopp's side host Genk at Anfield.
Liverpool signed Brazil international Alisson from AS Roma on Thursday, smashing the world record for a goalkeeper in a deal worth up to 72.5 million euros ($84 million).
A summary of Saturday's action in the Premier League.
Wolves took the lead a minute later when Rayan Ait-Nouri's cross found Sarabia poorly marked in the middle of the box. The Spain international headed past Grbic to score his third goal in the campaign.
Roma didn't expect that Salah and Liverpool would stand between them and a place in the Champions League final.
Al-Hilal wrapped up a place in the Asian Champions League semi-finals as a 2-0 win over fellow Saudi Pro League side Al-Ittihad in Jeddah on Tuesday completed a 4-0 aggregate success and set a world record with their 28th straight win in all competitions.
Everton have filed an appeal against their 10-point deduction by the Premier League, the club said on Friday.
Juergen Klopp expressed disappointment at the officiating and his Liverpool side's poor finishing in the second leg of their League Cup semi-final at Anfield on Wednesday after Southampton claimed a late 1-0 victory to book a spot in the final.
The Reds' manager believes his team has the ability to finish in the top four and qualify for Champions League next season.
And Liverpool fans could not keep their calm as they made their joy known on social media sites.
Liverpool beat Leicester City 2-1 to post a fourth successive Premier League victory but conceded their first goal of the season after a nightmare mistake by new keeper Alisson.
Barcelona have been haunted by their recent failures in the Champions League and there are concerning signs for the Catalans as they prepare for Tuesday's last-16 first leg away to Olympique Lyonnais.
Good news was in short supply for Manchester United manager Jose Mourinho on Saturday as his side were thrashed 4-1 by Liverpool in the International Champions Cup and he confirmed Serbian midfielder Nemanja Matic will miss the start of the season.
Images from Saturday's action in the English Premier League.
Strict rules govern transfers, with clubs able to negotiate terms with players only if they are out of contract or once a fee has been agreed with the seller. The laws also govern indirect approaches via an intermediary.
Arsenal eased past second-tier Reading 2-0 with Alex Oxlade-Chamberlain scoring twice, while Newcastle United romped to a 6-0 win over 10-man Preston North End and Hull City won 2-1 at Bristol City.
Whatever happens, there will be no shortage of drama between now and May 12.
Liverpool manager Juergen Klopp says his team can compensate for their lack of Champions League final experience by showing greater desire and work-rate than Real Madrid in Saturday's clash in Kiev.
While Lyon rallied to hold Leipzig 2-2 after a Memphis Depay strike, it was the English clubs who made a mark on the night. Liverpool and Chelsea moved into the last 16 with contrasting wins and now all four English clubs, including Manchester City and Tottenham Hotspur, are in the Champions League knockouts this season.
Liverpool reached the Champions League final after riding their luck to contain a fired-up AS Roma in a 4-2 defeat at the Stadio Olimpico on Wednesday that sent them through 7-6 on aggregate.
Teenager Curtis Jones grabbed a 58th-minute winner as an injury-hit Liverpool qualified for the knockout stage of the Champions League with a 1-0 win over Ajax Amsterdam at Anfield on Tuesday.
There is often nothing more dangerous than a striker with a point to prove as Daniel Sturridge and Divock Origi showed in Liverpool's League Cup mauling of Southampton on Wednesday.
Barcelona coach Ronald Koeman has joined other managers in lashing out at the packed European football calendar after deciding to rest Lionel Messi for Tuesday's Champions League trip to Dynamo Kyiv.
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