It's the eighth time Messi has scored a hat-trick in he Champions League 42nd since he made his first team debut for the club.
Messi marked other milestones too.
The first goal of the 2018-19 Champions League group stage was a stunner from one of the all-time greats.
He has now scored seven goals in six games this season, as well as going close with another free-kick in the second half and creating good chances for Suarez and Roberto.
"It's a blessing to have Messi in your side and a problem for anyone that doesn't [have him]", Barca coach Valverde said in his postgame news conference.
"Messi is a man of his word", wrote Marca.
Marca described it as a "Maradonian touch".More news: IRS grants tax extensions to Hurricane Florence victims
"It's time to win the Champions League", said Messi. "I hope he goes on and on like this forever and scores many more goals".
The Argentine inspired the Catalans to an easy win against the Dutch giants at the Camp Nou.
Ousmane Dembele also bagged a goal, his fifth of the season, as Blaugrana made the flawless start to their European campaign.
Goals from Garry Rodrigues, Eren Derdiyok and Selcuk Inan helped Galatasaray to a comfortable 3-0 win over Lokomotiv Moscow in Istanbul, despite the late sending-off of midfielder Badou N'Diaye. First Dembele, after leaving two PSV midfielders for dead with a dragback, unleashed into the bottom corner. "We should have stopped him", he said. He has never scored more in a year. "He was dazzling", wrote AS.
In the other Group C match, Red Star Belgrade held Napoli to a scoreless draw at home.
Barcelona's all-time top scorer opened the bank with a phenomenal free-kick at the 32nd minute.
Against PSV, FCB won 4-0, another convincing performance from a team that has looked definitively like the best in Spain.
Valverde's men famously capitulated in the second half of the away leg against the Giallorossi but, having significantly bolstered their armoury over the summer, another result like that seems unlikely.