Goals from Dirk Kuyt and Xabi Alonso helped Liverpool secure a 2-0 home win.
Striker Costa ends second spell at Atletico Madrid.
Brazilian forward Julio Baptista scored four goals as Arsenal thumped Liverpool 6-3 in their League Cup quarter-final.
Arsenal knock out Cup holders Liverpool There were no such problems for Chelsea, however, as a Frank Lampard hat-trick helped them thump Macclesfield in the FA Cup.
Tottenham manager Mauricio Pochettino is readying his squad for a final tilt at the Premier League title, aware that he can ask his players to remain focused on one game at a time when a bulk of them return from the international break.
Liverpool manager Juergen Klopp acknowledged that they would have to defend better in the visit to Stamford Bridge to stop Eden Hazard reprising the sort of solo brilliance that propelled Chelsea to victory at Anfield on Wednesday.
The new deal will keep the Spaniard at Anfield for another four years until 2010.
Liverpool manager Brendan Rodgers will speak to striker Raheem Sterling after footage emerged of the England international apparently taking nitrous oxide.
According to The Sun, the US owners of the Anfield giants, Fenway Sports Group, have become disillusioned with the club, as they have not managed to reach the top of the Premier League.
The English champions will go through to the last 16 of the Champions League if they overcome Real Betis in Seville.
Manager Brendan Rodgers has ended whatever honeymoon period new striker Mario Balotelli was enjoying at Anfield, telling him he needs to start scoring goals after finding the net just once in seven appearances for Liverpool.
England scored a 2-1 comeback win over Uruguay in a friendly at Anfield
"The situation is very simple. The club has agreed with another for a deal for Raheem to be transferred and subject to medical that will go through," Rodgers said.
Frank de Boer ruled out accepting the Tottenham Hotspur job again on Friday, saying he was satisfied with his post at Ajax Amsterdam.
Ole Gunnar Solskjaer continued his dream start as Manchester United caretaker manager with a 3-1 Premier League win over Huddersfield on his Old Trafford debut in the dugout on Wednesday.
Liverpool captain Steven Gerrard will be sidelined for at least four weeks with the hamstring injury he picked up last weekend, manager Brendan Rodgers said on Thursday.
The Spanish striker completed his nine million euros transfer from Real Madrid to Liverpool after passing a medical.
News of all that's transpired on and off the football field
Liverpool, five-times European champions and last season's Champions League runners-up, eased past the Bavarians and into the quarter-finals with two goals from Sadio Mane and one from Virgil van Dijk after a goalless first leg at Anfield.
Holders and Premier League leaders Arsenal will be at home to Championship (second tier) Burnley in the fourth round of the FA Cup after most of the big guns avoided top flight opponents in Monday's draw.
United beat Leicester City at Old Trafford in the opening game of the season but Spurs present a very different quality of opponent.
AC Milan scored a 4-0 win over Fenerbahce with Andrei Shevchenko scoring all four goals in their Champions League Group E victory.
Mario Balotelli enjoyed a rare moment of ecstasy in a Liverpool shirt by inspiring a 2-1 win over Swansea City in the League Cup on Tuesday while Chelsea had to call for reinforcements to see off Shrewsbury Town.
Liverpool turn to the relative calm of their Champions League defence with a place in the last 16 at stake on Tuesday.
Liverpool survived a late spell of Everton pressure to stretch their record unbeaten run in Merseyside derbies to 17 games
News of all that's transpired on and off the football field
Liverpool manager Juergen Klopp is not about to change his team's trademark style against Champions League quarter-final opponent
Sterling was jeered by some supporters, while others shouted "Stay, Raheem" as he picked up the accolade a day after British media reported that the 20-year-old was poised to request a transfer.
Liverpool ended their 85-year Manchester United jinx when a first-half Peter Crouch header gave them a 1-0 win in the FA Cup fifth round.
Liverpool have not beaten Manchester United in the FA Cup for 85 years
The Senegal international, 26, joined Liverpool from Southampton in 2016 for 34 million pounds ($43.8 million) and has scored 40 goals in 89 games for the Anfield club.
Striker Robbie Fowler has rejoined Liverpool on a free transfer until the end of the season
