Pogacar, 26, was riding his debut Paris-Roubaix, yet it was no surprise the world champion sparked the decisive move in the race.
The three-time Tour de France winner attacked a group of favourites at the front with 71km to go and only Van der Poel, 30, was able to counter.
The Dutchman stalled their progress to allow Alpecin-Deceuninck team-mate Jasper Philipsen to join before Philipsen slipped back 23km later.
From there it seemed likely to come down to which rider had the most power left, but Van der Poel’s superb bike-handling skills and Paris-Roubaix experience proved vital, with Pogacar taking the corner too quickly and sliding into the mud.
While not a serious crash, it was too much of an advantage to cede, even with Van der Poel later suffering a puncture, and the Dutchman crossed the line just before Pogacar entered the velodrome, winning by one minute and 18 seconds.
Denmark’s Mads Pedersen won the sprint for third, holding off Belgians Wout van Aert and Florian Vermeersch. Britain’s Fred Wright finished ninth.
“We all know what an incredible champion Tadej is – he was in his first Roubaix, it doesn’t surprise me but it’s also not normal, he’s an exceptional talent,” added Van der Poel.
“It would have been the two of us going to the velodome if he didn’t make the mistake so I’ll guess we’ll see him back next year to take his revenge.”
Van der Poel now has eight wins across the five oldest and most prestigious ‘Monument’ races in men’s cycling, drawing level again with Pogacar after the Slovenian claimed the Tour of Flanders last week.
Only five riders have won more Monuments: Eddy Mercx (19), Roger de Vlaeminck (11) and Costante Girardengo, Fausto Coppi and Sean Kelly (nine).
Van der Poel and Pogacar have won all three Monuments between them this year, though Pogacar will be heavily favoured in the remaining two – Liege-Bastogne-Liege and Il Lombardia.
Pogacar has two Liege-Bastogne-Liege titles and has won the last four editions of Il Lombardia, with his climbing ability well suited to the hillier terrain of both races.
