The Olympic champion was the highest-profile casualty on day one, losing to a player making her main-draw debut in a WTA 1000 event.
Only the second Danish player to appear in Doha, after former world No.1 Caroline Wozniacki, the 19-year-old Tauson came from 4-2 down to take the opening set, before Bencic levelled up.
But Tauson was not to be denied. The world No.33 gained a crucial break in game four of the decider, before winning 12 of the last 13 points on her serve to set up a second-round clash with No.3 seed Paula Badosa.
"It was a very tough match," Tauson, who hit seven aces, said. "I know Belinda is playing very well, so I had to play my best today, and I did sometimes.
"I have my confidence, but every match is really tough at this level, so I'm doing my best."
Two-time champion Victoria Azarenka survived a scare to scrape past Kazakhstan's Yulia Putintseva 5-7 6-2 7-5.
The No.12 seed, who is aiming to become the first player to lift this trophy three times, recovered from two breaks down in the decider before rescuing match point at 5-4 down and eventually scrambling over the line.
Amanda Anisimova, who defeated Bencic and Naomi Osaka at the Australian Open, built on her early-season momentum by seeing off Beatriz Haddad Maia 7-5 6-4.
No.9 seed Jessie Pegula, a quarter-finalist in the season's opening grand slam in Melbourne, beat doubles world No.1 Katerina Siniakova in straight sets, while former Australian Open champion Sofia Kenin lost in three sets to fellow American Ann Li.