The victory was briefly soured when France defender Umtiti was sent off in the 79th minute for a second yellow card.
Messi will be looking to help lead Barcelona to its first Champions League title since 2015. The Catalan club has won 25 games and drawn only two during the streak.
"It's time to win the Champions League", said Messi.
After missing the majority of last season injured following his €105 million move from Borussia Dortmund, the French forward has scored the key goal in narrow wins over Sevilla, Real Valladolid and Real Sociedad.
The home side started the second half strongly, and just before the hour mark had a Mo Salah goal ruled out for a foul by Sturridge on the PSG goalkeeper.
Messi sealed his hat trick and confirmed Barcelona's rout with goals in the 77th and 87th minutes.
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Dembele's season started strongly with the victor in Barcelona's Spanish Super Cup clash with Sevilla. Two minutes later, Messi got his second goal turning Ivan Rakitic's dinked pass into the near corner with one flick of his boot. Curiously, it's the third hat-trick that he has scored in first game of the group phase.
"Barcelona fight for all the competitions but the Champions League is the biggest and we are going to go for it". The Dutch club gained some momentum halfway through second half but wasn't able to capitalize on its brief dominance.
"He makes the extraordinary look ordinary", said Barcelona's coach Ernesto Valverde.
Messi and Barcelona are looking to break Spanish rival Real's three-year stranglehold on the Champions League trophy this season. 'The injury made the first year hard for him.
"It's not easy, this is the Champions League and also our first match against a team who have started well in their domestic league, but it's been a game that worked out for us", Rakitic told reporters.
Regarded by many as the greatest footballer of our generation, arguably, Messi has done things on the pitch that other footballers can only dream of, if not execute on a video game.