Purvis, second-last to perform, put together a solid routine after his double twist double opener and nailed his landings to move into the bronze medal position with 15.400 points.
It edged the 22-year-old Scot ahead of Brazil's Diego Hypolito into a podium position after qualifying in fifth place, with only one gymnast to come. But Japan's Kenzo Shirai, the best in qualifying, produced an exceptional performance to achieve 16.000 and take the gold and move Purvis down the rankings, although he still finished top European.
The 17-year-old brought the crowd to their feet with his routine which had a high degree of difficulty.
American Jake Dalton took silver with 15.600 points and Japan's Kohei Uchimura, who won his fourth successive all-around world title on Thursday, took bronze (15.500 points), just ahead of Purvis.
The Scot, a real hope for a Commonwealth Games medal in Glasgow next year, finished ahead of German Fabian Hambuechen and Hypolito, who shared fifth place.
"Going into it I just wanted to enjoy it and then as it progressed I thought, 'I can get quite close here' and was really happy to nail my routine, my best of the championships," Purvis said.
"I've been strong all week and I'm really satisfied and proud of my World Championships. I knew that Shirai was last up so enjoyed being in third for a while, but knew realistically he is at another level."
Purvis made a late decision on which routine to use after discussing it with his coach Andrei Popov on Friday evening, but was happy with the way it went.
"I was a bit tired after the all-around finals, but just wanted to put in a strong performance," he said. "I knew I had a medal chance, but I was happy with the way everything went."
Purvis, born in Liverpool but who qualifies as a Scot as his mother Denise hails from Dundee, won a bronze at the World Championships in Rotterdam three years ago, but has enjoyed an excellent week, finishing seventh in the all-around final on Thursday night and showing he can hold his own at the highest level. By finishing in the top eight in the all-around final, he qualifies for the World Cup at Glasgow's Emirates Arena in December.
Whitlock, the 20-year-old Olympic double bronze medallist, completed the most difficult pommel horse routine in the world - with a 7.2 difficulty tariff - to score 15.633 and go one better than his London 2012 third place to finish behind Japan's Kohei Kameyama, who won gold with 15.833.
"Some of the bits were scrappy, but I'm just happy to get through the routine," said Whitlock, who was tied for second place with Mexico's Daniel Corral Barron and finished fourth in Thursday's all-around final. The Hemel Hempstead-born gymnast hopes his results from this competition mark an upward curve in his form as he continues on the road to the Rio 2016 Olympic Games.
He added: "I'm looking forward to the competitions ahead. Hopefully I will be able to qualify for them and I'm looking forward to those experiences."
Ruby Harrold and Beckie Downie finished seventh and eighth in the uneven bars final as China's Huang Huidan took gold.
McKayla Maroney won women's vault gold with a score of 15.724 ahead of fellow American Simone Biles.