The team from the canton of Valais dominated their opponents, who have just come off winning the Super League championship, and it did so on Basel's home ground, Saint Jakob-Park.
Valais, egged on by a large contingent of supporters in the stands, were superior in all the domains, “magic quite simply”, state broadcaster RTS reported.
Moussa Konaté opened the scoring for Sion in the 18th minute.
The second goal came five minutes into the second half when Edimilson Fernandes made it 2-0.
Carlitos notched the third goal in the 60th minute with a header.
The victory for Sion is the 13th out of 13 Swiss cup finals that it has competed in, matching the number of stars on the flag of the canton of Valais, which is this year celebrating its 200th anniversary in the Swiss confederation.
The win automatically qualifies Sion to play next year in the Europa Cup.
Fans gathered in Sion to greet the team when it arrived from Basel on Sunday night.
Red and white bunting, the team's colours, adorned streets in the city centre, where supported watched the game on a large video screen.
In Basel, police said after the match ended they fired rubber bullets to quell clashes between supporters from the two teams in various areas of the city centre, including the underpass of the St. Jakobsstrasse.
One man was injured and taken to hospital, while two people were arrested and two officers were treated for injuries, police said.
Founded in 1925, the Swiss Cup is similar to England's FA Cup, with a round-robin playoff system involving 64 professional and amateur teams.