India was put in to bat after losing the toss at The County Ground but looked impressive from the start as Mandhana, Raj and Poonam Raut did most of the damage in a total of 281 for three from 50 overs.
India women now take on West Indies in their second match on June 29.
The captain of India's women's cricket team caught fans' attention during a World Cup game on Saturday by calmly reading a book as she waited to bat.
'It's about how we react now, ' she said. But just when the going seemed good, Mandhana was sent packing by off-spinner Heather Knight.
The 20-year-old Mandhana struck 11 fours and two sixes off 72 deliveries while Raut assumed the sheet anchor role, facing 134 balls and counting seven fours and a six.
She joined Knight to rebuild after the England top order failed - among them Sarah Taylor, returning to an global cricket after a year away with an anxiety-related issue, who made 22.
In reply, England were bundled out for 246 runs. That was our plan - to get off to a good start. "I am always into Kindle or books as I don't get those jitters even before getting into batting".
India captain Mithali Raj, taking part in her fifth World Cup, said: "It was a brilliant partnership between our openers".
"After a disappointing 2013 World Cup at home, they've backed themselves and played with confidence, and Mithali Raj has clearly done her homework on England".
It is India's first win over England in five years as they had all lost each of the six past matches to the hosts.
In the other match, New Zealand defeated Sri Lanka by nine wickets in Bristol.
Chasing 189, Suzie Bates (106 not out) and Amy Satterthwaite (78 not out) produced some fine innings to overhaul the target in 38th over. Mithali has been called Lady Sachin for her big individual scores, which have often tipped the scales in favour of her team, given them the first ever entry into the final of the World Cup in 2005 and clinched the World T20 bilaterial series against Australia 2-0.
After the top three players - Nipuni Hansika (31), Chamari Polgampola (49) and Chamari Atapattu (53) - departed, only one Sri Lankan made double figures as Huddleston ripped through the tail.