Messi, however, continued his dream run by scoring goals at 77 and 87 minute.
After leaving the pitch the five-time Champions League victor, whose stoppage-time penalty for Real Madrid knocked Juve out in the quarter-finals last season, waited by the tunnel entrance, seemingly on the verge of tears. "It was a magisterial free kick", Valverde added.
If the World Cup proved anything, however, it is that Messi can not do it alone and Barcelona Messi will need others to shine if they are to end their Champions League hiatus.
Barca's Samuel Umtiti was dismissed in the 79th minute for a second yellow card.
With his latest hat-trick, he now leads his great rival Cristiano Ronaldo 63-60 in total group-stage goals.
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.
Barcelona will also wonder how they would have made the breakthrough without the individual genius of Messi.
Suarez then teed up Messi for his third as the Argentina global calmly slotted home from 10 yards away to put the result even further out of reach.
With a turbulent debut season, blighted by injury, behind him, Dembele looks like a player back on track. "We haven't won it in a few years but we have the players and we are hungry to go out there and win it".
PSV are the 30th team Messi has scored against in the Champions League, . "This was typical of the way Messi has been playing the last two, three years". "He is the best player in the world, [the best] in the history of the game". "But he makes the extraordinary routine".
More than just three points for Inter Milan: It looked like near certainty that Tottenham would get three points after Christian Eriksen's opener and Inter's inability to do anything in attack.
Breel Embolo gave the German side a second-half lead, firing past Iker Casillas, who has now played in the Champions League in a record 20 separate seasons.