Only Scot Alain Baxter has finished higher in a World Cup slalom, when he was fourth in Are, Sweden, in 2001.
Konrad Bartelski holds the best British alpine result after coming second in a downhill race in Val Gardena, Italy, 35 years ago.
Ryding, from Lancashire, had never previously finished in the top 10.
He told BBC Sport: "I am delighted. I am sure no-one expected a Brit in the top six!
"I have been skiing well and performing in training but this was better than I would have expected.
"I really had to focus before the second run and give it everything to stay up the rankings. I was aiming to get a solid top 15 so sixth is amazing."
Ryding does not receive UK Sport funding as he builds up to the 2018 Winter Olympics, which will be the 29-year-old's third Games.
In Finland, Ryding was fourth after the first run of two on a foggy Levi course but a couple of errors cost him a podium place.
Austrian world number one Marcel Hirscher claimed his 40th World Cup victory ahead of compatriot Michael Matt, with Italy's Manfred Molgg third.
Ryding's result capped a successful weekend for British winter sports as snowboarders Aimee Fuller and Billy Morgan finished fourth and seventh respectively at a World Cup big air in Milan.
Big air will make its debut in the Pyeongchang Olympics in February 2018.
Dave Ryding recorded a best British alpine result for 15 years as he finished a career-high sixth at a World Cup slalom in Finland.