Chasing 282 to win, England failed to build partnership with middle-order batswoman Fran Wilson (81 off 102) emerging as the highest scorer.
The openers ran well in a stand of 144 that was only ended when Mandhana hit Heather Knight's off-spin to mid-wicket.
They came unstuck against an inspired India side who are two places below them in the world rankings, in fourth, but showed more intent with the bat, more discipline with the ball and some electricity in the field.
West Indies also come into the tournament with plenty of confidence having lifted the ICC Women's World Twenty20 just past year, as part of a hat-trick that included the ICC World Twenty20 won by the men's team and the ICC U19 Cricket World Cup taken home by their colts.
Classy captain Raj arrived to share 78 for the second wicket with Raut, who eventually holed out on the leg-side boundary off Danielle Hazell.
Wilson, in only her seventh one-day global, may not have been playing but for Lauren Winfield being ruled out with a wrist injury. Her run out proved the turning point since England was cruising at that stage. That was our plan - to get off to a good start.
"I think I enjoyed batting with Punam. We didn't bowl the way we wanted to and India put the pressure on us".
Meet Mithali Raj, the captain of the Indian women's cricket team from 2005 to 2008 and from 2012 till present, a total of eight years, and under her the team has blossomed into one which is winning matches at home and away. "I realised the wicket didn't assist the bowlers but the spinners bowled in the right areas", Raj said after the match. She smashed the half ton at the ICC Women's World Cup, against England in Derby.
At Bristol, New Zealand defeated Sri Lanka by nine wickets. 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.
Chamari Jayangani had earlier top-scored with 53 while opener Chamari Kumarihami got 49, but seamer Holly Huddleston's five-wicket haul limited the Sri Lankans to 188 for nine off their 50 overs.