Dhar made 57 from 130 balls and Sharma hit 50 from 95 balls to save their team after the top order collapsed with scorecard reading just 74 for six at one stage in their chase of 271 in 70 overs Friday.
The duo put up 77 for the seventh wicket and at one stage raised a feeble hope of pulling off a win after their mainstays Monica Sumra, captain Mithali Raj and Anjum Chopra failed to score.
The visitors made 187 for eight in their second innings after England declared at 345 for eight and fell short of 83 runs to settle for their 10th draw in 11 meetings against the hosts.
Dhar, who struck 10 boundaries, was finally dismissed by Caroline Atkins.
Jenny Gunn, Laura Marsh and Laura Newton took two wickets each.
The second and final Test is scheduled at Taunton at the end of the month after the conclusion of the five-match One-day series.
England 1st innings: 223 all out; Caroline Atkins 73; Jhulian Goswami 3 for 46
India 1st innings: 298 all out; Jhulian Goswami 69; H Kala 69; Laura Newton 3 for 57
England 2nd innings: SC Taylor 115; Laura Newton 85; P Dimri 3 for 75
India 2nd innings: 187 for eight; Rumeli Dhar 57; Amita Sharma 50; Laura Marsh 3 for 44