Published: 2017-10-19 09:45:35 BdST
Gerard Pique was sent off two minutes before halftime for two bookings, one for tugging an opponent's shirt and the other for slapping the ball into the net, but Barca were barely threatened by a hapless Olympiakos side who are badly under-performing in their Greek title defence.
The Athens club, who won three league titles and two domestic Cups under current Barca coach Ernesto Valverde, are fifth in the standings in the Greek top flight and have lost all their games in the Champions League.
Barca took the lead in the 18th minute at a rainswept and sparsely attended Nou Camp when visiting defender Dimitrios Nikolaou turned Gerard Deulofeu's cross into his own net.
It was the fifth own goal Barca have benefited from this season in all competitions.
Messi struck his landmark goal in Europe from a free kick in the 61st minute to double Barca's lead. It was his 97th goal in the Champions League. His other three came in the UEFA Super Cup.
"I hope he can get to 200 by the end of this season," joked Valverde after the Argentine took his tally to 15 goals in all competitions less than two months into the campaign.
Champions League - FC Barcelona vs Olympiacos - Camp Nou, Barcelona, Spain - Oct 18, 2017 Barcelona’s Lucas Digne celebrates scoring their third goal with Paulinho, Lionel Messi and Luis Suarez. Reuters
Barca president Josep Maria Bartomeu also paid tribute to Messi.
"I would give him every prize in the world, he has been the best player of all time for many years now," he said.
"He has made this era sublime. We see him as the leader of our team, he contributes so much and we all never tire of enjoying him."
Messi then carved his way through the hapless visiting defence moments later to tee up France left back Lucas Digne to score his first goal in the Champions League for Barca.
Greek defender Nikolaou, 19, scored for a second time but first for his side when he headed home from a corner to give Olympiakos a scantly-deserved consolation goal in the 89th minute.
The unbeaten Barca top Champions League Group D with nine points. A win at Olympiakos in two weeks will seal their place in the last 16.
"We are feeling very good in the league and Champions League and we have to keep going like this," Barca's Sergi Roberto told reporters after his side recorded a 10th victory in 11 games.
"We are frustrated to have conceded a goal although by then the game was almost over."