England made 177 for seven from their 20 overs, with Sciver top-scoring with 55 from just 27 deliveries and Jones contributing 43 as the pair shared a crucial 78-run partnership in just seven overs.
India looked threatening at the start of their chase following the early loss of Shafali Verma, with Smriti Mandhana hitting 29 from 17 deliveries, but they lost wickets steadily and were far behind the required rate on 54 for three when the rain came down.
England’s victory sees them win the first of three T20 matches in the multi-format series, with the teams meeting again in Hove on Sunday.
The hosts were sent into bat after India won the toss and despite not finding the boundary regularly early on – scoring just one four in the opening four overs – they made a solid start and were 37 without loss after the opening five overs.
Tammy Beaumont and Danni Wyatt’s opening stand was finally ended with the score on 56, and when captain Heather Knight departed in the 12th over, England were 85 for three.
But Sciver and Jones joined forces to help guide their side towards an impressive total.
Sciver brought up her half-century off just 24 balls, with just two dots, but towards the end of England’s innings India fought back with some brilliant bits of fielding to slow the home side’s progress.
It took a well-timed low diving catch from Harmanpreet Kaur to dismiss Sciver with less than two overs remaining.
However the best was yet to come as Jones, who had struck two sixes and four fours, fell victim to a brilliant catch later in the same Shikha Pandey over.
Harleen Deol got two hands to the ball on the boundary rope, parried it up before jumping back onto the field to take the catch to dismiss Jones.
Just one ball later, a good piece of work from wicketkeeper Richa Ghosh saw Sophia Dunkley stumped, and there was another fine piece of glovework from Ghosh to run out Katherine Brunt off the final ball of the innings.
Brunt was quickly into her stride as India replied though, striking with just the second ball of their chase to dismiss the dangerous teenager Verma, who found herself trapped on the crease.
Mandhana was caught by Sophie Ecclestone off the bowling of Sciver and captain Kaur lasted just two balls before falling to Sarah Glenn.
India had little time to recover from those setbacks as rain stopped play in the ninth over, with England decisively ahead on DLS.