That made it 2-0 on the night in the first leg of the Champions League semi-final.
Goal 600, however, was quite special. The Liverpool midfielder Fabinho bodychecked Messi as he tried to collect a pass from Suarez.
The free-kick was given 30 metres from the goal. Four Liverpool playes lined up in a wall to protect the right hand side of Alisson's goal.
Messi's left foot rendered them redundant. The ball flew to their left but straightened as it arrowed into the top right hand corner past Alisson's forlorn glove. 3-0.
As the Camp Nou faithful hailed their hero, the Liverpool boss Jurgen Klopp smiled wryly. They call it Galgenhumor in his native Germany - a brief moment of mirth at the point of execution.
"He's unstoppable," said Klopp of Messi. "Obviously, we couldn't defend the free-kick. What a strike. He's a world-class player, but I knew that already. It's no surprise."
Liverpool old boy Suarez scored Barcelona's first goal after 26 minutes. It was his 27th strike of the season. Messi's brace took his tally to 47.
Their Liverpool counterparts - Sadio Mané and Mo Salah - threatened throughout but could not capitalise on the chances. Salah hit the post shortly after Barcelona's third goal.
"I said to the boys that I was proud of how we played against a side like this," Klopp said. "If you lose away, it's not a massive problem as long as you score a goal. That's our problem. It doesn't make our life any easier."
Barcelona are now favourites to advance to their first Champions League final since they claimed the crown in 2015.
However, after losing 3-0 at Roma in the second leg of last season's quarter-finals, coach Ernesto Valverde warned that his side would be on their guard at Anfield on 7 May.
"They can still make us suffer," said Valverde. "This result is good but the tie is possible."