The 31-year-old suffered a puncture as did Pogacar and great rival Mathieu van der Poel, who had two mechanicals and could not contest the final sprint, which went in Van Aert's favour for his second title in one of the five Monument classics after his Milan-Sanremo victory in 2020.
Van Aert, who had been plagued by bad luck in the Queen of the Classics, thwarted Pogacar's attacks on the cobbles and wrapped it up with a trademark burst of speed in the final straight on the Roubaix Velodrome.
Fellow Belgian Jasper Stuyven took third place, 13 seconds behind.
Van der Poel, seeking a fourth consecutive win in the race, finished fourth despite losing more than two minutes following a mechanical issue on a tricky cobbled sector of the course.
"It's everything to me, it's been a goal since I first did this race. I stopped believing a lot of times but I would start believing again the next day," said Van Aert, who had suffered repeated punctures and crashes in the Flanders classics.
The Visma-Lease a Bike rider dedicated his title to former teammate Michael Goolaerts, who died in 2018 after suffering a cardiac arrest during the race.
"Arriving for a final sprint with the world champion and beating him in a sprint is very special," Van Aert added.
Pogacar, who won the first two Monument classics of the season in the Milan-Sanremo and the Tour of Flanders, was looking to become the first Tour de France champion to prevail in the "Hell of the North" since France's Bernard Hinault in 1981.
German Franziska Koch won the women's race in a sprint against Dutch great Marianne Vos, who had skipped the Tour of Flanders last Sunday following the recent death of her father.
Koch prevailed although she was facing a two-pronged challenge from Visma-Lease a Bike, who had Vos and Tour de France champions Pauline Ferrand-Prevot combining in the finale.