Off-spinner Charlie Dean then cut through the India middle order, taking the wickets of Harmanpreet Kaur — a century-maker against the West Indies — and of Sneh Rana and Pooja Vastrakar as India struggled to mount a recovery.
Two run-outs contributed to India's problems and they were 86-7 when Vastrakar was out.
Sophie Ecclestone claimed the crucial wicket of Smitri Mandhana (35), who made 123 against the West Indies, and Dean returned to end the innings and return 4-23 from 8.2 overs.
"To go out there and perform was pretty special," Dean said. "We talked about keeping it simple and we pulled it off today. It's not the World Cup we have hoped for but the girls are really in this. This will keep us going in this World Cup."
Shrubsole followed the policy of fuller lengths to achieve a personal milestone. She bowled across the left-handed Yastika Bhatia in the fourth over and brought the ball back enough to beat the bat and claim her 100th wicket in one-day internationals.
India's opening bowler Jhulan Goswami later claimed her 250th ODI wicket.
England's fielding picked up a notch in this match. Dropped catches have been costly recently but none went down on Wednesday and sharp fielding brought the run out of Deepti Sharma in the eight over. Having been unable to score from 10 balls, Sharma drove just wide of mid-off and ran but Kate Cross pounced on the ball and, with three stumps to aim at, hit directly with Sharma well short of her ground.
England began their chase for India's modest total with the chance of improving their net run-rate, which might play a role later in determining the semifinalists.
But they were set back by the loss of Danni Wyatt in the second over and Tammy Beaumont in the third. England's opening partnerships at the tournament before Wednesday's match have averaged only 11 runs and remain a cause of concern.
Sciver's 46-ball innings set England back on track and Knight reached her 23rd ODI half century from 66 balls.
England's run rate increased with cameos from Amy Jones, whose 10 runs included a six, and Sophia Dunkley whose 17 included four boundaries. But two late wickets added again to the drama.
"To win this, we needed that," Knight said. "It was a little bit scrappy and we lost more wickets than we would have liked but to get over the line is really nice for us and hopefully the start of a momentum shift."
India now also face pressure to reach the top four and face unbeaten Australia in their next match on Saturday.
Kaur sustained a knee injury to cause some concern for the India team.
