The lean paceman came up with a burst of five for 14, his third fifer in Test cricket, to help India bowl out the hosts for 225 in their first innings and take a lead of 128 runs.
The visitors then raced to 31 for no loss at tea to take their overall lead to 159.
After lunch, India made good use of their second new ball that was only three overs old. Kumar went on to bowl unchanged for 10.4 overs and completely changed the direction of the game with his consistent line and length.
The 200-mark had come up for them in the 87th over, and thereafter the havoc began. Kumar first removed Jermaine Blackwood (20), caught at second slip by skipper Virat Kohli, thus ending his 67-run partnership with Marlon Samuels (48.
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Samuels was the next to go, two overs later, set up beautifully with a slew of outswingers and playing off an inswinging delivery.
Ravindra Jadeja (1/27) then removed Roston Chase (2) at the other end, and the hosts were really struggling at this juncture with three wickets down for just three runs.
R Ashwin (2-52) then accounted for Miguel Cummins (0) as Kumar searched for his fifth wicket even as Shane Dowrich (18) was dropped twice in successive overs, first in slips by KL Rahul and then at short midwicket by Rohit Sharma, who dropped a skier.
Finally, in the 104th over of the innings, the medium pacer celebrated his achievement with Dowrich trapped before the wickets as the West Indies lost six wickets for 23 runs.
Earlier, India removed both overnight batsmen but the West Indies continued to defy the visiting bowlers.
Samuels (39) and Jermaine Blackwood (20) added 59 runs for the undefeated fourth wicket in an extended morning session as rain stayed away today even as wet outfield prevented an early start.
Starting at overnight score of 107 for 1, Brathwaite (64) and Bravo (29) took to the crease. The conditions were a bit like the first morning with the pitch feeling fresh.
Jadeja and Mohammed Shami got things underway, with Ishant Sharma soon taking over from the spinner.
The umpires checked for no ball and it seemed that Ishant had overstepped marginally but the third umpire ruled in the bowler's favour.
It became a double blow for the West Indies, as R Ashwin (1*/51) removed Brathwaite at the other end three overs later, the batsman gloving down leg side and caught behind. It brought the proceedings to life, with only a small chance of result possible when play had started.
The fielder was later hit on the helmet grille, to go with a hit at the back of the helmet on day two, but he was safe and continued fielding, as the 150-mark came up in the 63rd over.
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Mishra, who took just one wicket and looked unimpressive in the England first innings, broke the ballooning stand by sending back Hameed in his very first over of the morning late in the session and added the prize scalp of Joe Root in his next over to bring some life to the dreary proceedings.
Replays were called to see if Mishra had over-stepped but he had landed his front foot inside the crease by the barest margin and Hameed, who played spin and pace with equal felicity, departed after having added 180 runs with his captain for the first wicket.
This was followed by the quick dismissal of Root who tried an almighty heave off Mishra's leg break and top-edged to wicketkeeper Wriddhiman Saha who took a comfortable catch.
Cook, overnight 46 and 106 at lunch, had 13 hits to the fence in his score of 130, before lofting Ashwin to long-off fielder Ravindra Jadeja and providing the off-spinner, who bowled economically by sticking to a leg side attack with a packed field, with his only wicket of the innings.
In all, the visiting team skipper had batted for just under five hours and faced 243 balls.
The trio of Indian spinners hardly got England in trouble in the second innings too.
Mishra erred in line and length after claiming the two quick wickets and was punished as gave away 60 runs in 13 overs.
Ravindra Jadeja too could not make much of an impression and ended up without adding to his tally of three first innings wickets.
Ashwin secured 1 for 63 to add to his costly two-wicket haul in the first innings.