Faced with a daunting chase, Afghanistan lost wickets at steady intervals in a feeble reply to finish day two on 113-5 and trailing by 451 runs at the New PCA Stadium in New Chandigarh.
Rahmat Shah was batting on 43 at stumps with Afghanistan needing a special knock from him on Monday.
Suthar claimed the wicket of opener Abdul Malik with just his fourth ball in Test cricket, before going on to dismiss Rahmanullah Gurbaz and Afsar Zazai, while giving away just 21 runs in his 15.5 overs.
"The wicket was offering some help, and my strength is getting the ball to turn," the 23-year-old Suthar said.
"So my aim was to extract as much turn as possible because the pitch was assisting. At the same time, I wanted to keep bowling in good areas and hit the ideal length consistently.
"Playing for India is a dream for everyone, and I am extremely happy to have received my cap."
Krishna was similarly impressive, breaching Sediqullah Atal's defence with a vicious delivery to claim his first Test wicket in India. He then dismissed Hashmatullah Shahidi, breaking up the Afghanistan captain's dangerous-looking partnership with Shah.
Earlier on Sunday, India resumed on 368-3 with captain Shubman Gill and Rishabh Pant looking to heap more misery on the visitors, but Afghanistan's bowlers did well to peg them back as Mohammad Saleem took a string of key wickets.
Gill (126) did not last long and was caught behind when Saleem got the ball to move away from him, with the fast bowler also dismissing Dhruv Jurel (19), Suthar (28) and Mohammed Siraj (22).
Saleem, 23, completed a six-wicket haul in just the second Test of his career, after having failed to pick up a wicket in his only previous match in the longest format.
Shahidi also did well to bait Pant (81) into swinging at a flighted ball, which the batter sent to long-off.
Despite the wickets, all-rounder Washington Sundar (52 not out) kept the scoreboard ticking for India and scored a composed half-century before Gill made the decision to declare.