Barnes, 21, sprinted over the line in her home county Northamptonshire behind winner Emma Johansson of Sweden and double Olympic champion Marianne Vos.
The British circuit race champion admitted she was a bag of nerves at the start of the 93.8km leg from Oundle, but was thrilled with her podium finish.
“I’ve been nervous since I knew we were coming to the race, so to get stage one out of the way with this result is great,” said Barnes, who rides for the UnitedHealthcare team.
“I didn’t expect to be in third place, so to do that and win two jerseys is great. It was one of my best-ever results. Now I have to decide which jersey I’m going to wear, which will be quite exciting.”
Barnes’s team-mate Sharon Laws claimed the Queen of the Mountains jersey and was blown away by the first day of the Tour.
“I didn’t think we’d see women’s cycling like this in Britain,” she said. “It’s bigger than any race I’ve been at.”
The five-stage Tour continues today from Hinckley to Bedford, with Barnes joined among the early front-runners by Briton’s Lucy Garner in sixth, and eighth-placed Armitstead, while track specialists King and Laura Trott also finished among the peloton.
STAGE ONE RESULTS
1 Emma Johansson (Sweden)
2 Marianne Vos (Holland)
3 Hannah Barnes (Great Britain)
4 Trixi Worrack (Germany)
5 Giorgia Bronzini (Italy)
6 Lucy Garner (Great Britain)
7 Elena Cecchini (Italy)