Carse trapped Karun Nair and captain Shubman Gill lbw in a crucial spell from the Pavilion End. Ben Stokes then bowled nightwatchman Akash Deep with the final ball of the day, leaving India 58-4, still 135 runs short of their target of 193.
Earlier, England were bowled out for 192 on a deteriorating pitch. Washington Sundar took 4-22, while Jasprit Bumrah claimed two wickets. England collapsed from a strong position, losing their last six wickets for 38 runs.
Harry Brook was bowled for 23 sweeping Akash Deep. Joe Root made 40 and Stokes 33, both falling to Sundar. In total, 12 England batters were bowled â a record for a home Test.
In Indiaâs reply, Jofra Archer dismissed Yashasvi Jaiswal for a duck. Chris Woakes dropped a return catch off KL Rahul when he was on 5.
England looked to be fading before Carseâs impact and Stokesâ final strike. The match is finely balanced heading into the final day.
Shoaib Bashir, who injured his finger on day three, is available to bowl but is likely to miss the final two Tests.
Third Rothesay Test, Lord's (day four of five)
England 387 (Root 104, Carse 56, Smith 51; Bumrah 5-74) & 192 (Root 40; Sundar 4-22)
India 387 (Rahul 100, Pant 74, Jadeja 72; Woakes 3-84) & 58-4 (Rahul 33*, Carse 2-11)
India need 135 runs to win