Liverpool overcame an early scare and came from a goal down to claim a 3-1 win at Austria's LASK Linz.
Bayern beat Borussia Moenchengladbach 3-0 on Saturday to bag their 13th win from 13 matches across all competitions this season.
Former Europa League winners Manchester United booked a quarter-final spot in the tournament after their second-string side laboured to a 2-1 home win over LASK Linz on Wednesday to seal a 7-1 aggregate rout of the Austrian side.
Manchester United took a big step towards the Europa League quarter-finals by thrashing Austrian side LASK Linz 5-0 on Thursday in their last 16 first-leg tie at an empty Linzer Stadium due to the coronavirus pandemic.
Leaders Porto suffered a shock 2-1 defeat at Famalicao as Portugal's Primeira Liga became the latest league to return to action following the coronavirus stoppage on Wednesday.
Klopp has led Liverpool to back-to-back Champions League finals but is adamant that their impressive recent track record does not make them the title favourites.
London sides Tottenham Hotspur and Arsenal made winning starts as did fellow Premier League club Leicester City.
IMAGES from the Europa League matches played on Thursday.
Rennes denied late Europa League equaliser due to little-known rule
Manchester United, Sevilla and Celtic led a parade of clubs with continental pedigree to book the first places in this season's Europa League knockout stage.
Images from Wednesday's Champions League matches, between Real Madrid and Paris St Germain, and Manchester City and Sporting CP
Arsenal suffered a shock Europa League exit to Olympiakos Piraeus on away goals after a 2-1 extra-time home defeat, while Manchester United and Wolverhampton Wanderers cruised into the last 16 with emphatic aggregate wins on Thursday.
Manchester United boss Ole Gunnar Solskjaer says the next step in his side's development would be securing a trophy as they look to cap off a promising season by winning the Europa League. United, who finished third in the Premier League, hold a 5-0 lead over Austrian side LASK Linz heading into the second leg of their last-16 tie which will be played without fans at Old Trafford on Wednesday.
Martial scored his 23rd goal for United in all competitions this season against LASK Linz in Wednesday's Europa League last-16 victory.
Seven-time European champions AC Milan hit back from two goals down against Celtic to claim a 4-2 victory that sent them into the Europa League knockout phase on Thursday as they were among 14 teams to progress in Europe's second-tier competition.
Barcelona's Champions League last-16 second leg match with Napoli scheduled for March 18 will take place without spectators due to fears over the spread of the coronavirus, a club spokesman said on Tuesday. The match will take place at Camp Nou after a 1-1 draw in Napoli, but no supporters will be allowed to enter the stadium.