Liverpool, playing in the Champions League's 'poorer cousin' for the second season running after exiting the premier competition at the group stage last term, beat Rabotniki in Macedonia while the Italians overcame Ireland's Shamrock Rovers.
The former European champions were among a smattering of familiar names in Thursday's 34 third qualifying round, first leg ties which featured matches from the island of Madeira, in the Atlantic ocean, to Novosibirsk, halfway across Russia.
Portugal's Maritimo hammered Bangor City 8-2 in the former match in Funchal while the latter saw Russian Cup runners-up Sibir Novosibirsk beat Apollon Limassol 1-0 to record their first-ever win in European club competition.
Senegal's Baba Diawara and Brazilians Tcho and Danilo all notched two goals apiece for Maritimo, who scored five times in the last 15 minutes.
"The victory is the first for Sibir in Europe, and I would like to congratulate the team and the city," said coach Igor Kriushenko.
However, coach Christian Gross was not pleased with his side who twice squandered the lead. "We have to play with more focus to make it easier for ourselves," he said.
There was crowd trouble in Romania when about 500 supporters of Croatia's Hajduk Split were banned from attending the match at Dinamo Bucharest following pre-game clashes.
Romania state news agency Agerpres said 25 Croat supporters and 25 Romanians clashed in front of a central shopping mall in Bucharest, with one supporter severely wounded. Dinamo won 3-1.
