After frustrating the Black Caps in the morning session by taking their overnight score to 316, India assumed control as they reduced the visitors to 7-128 before bad light forced an early end to play.
Wriddhiman Saha starred early on with an unbeaten half-century, but it was Kumar who swung the match firmly in India's favour by taking full advantage of an indifferent wicket to run through a struggling top order.
After removing a typically aggressive Martin Guptill early on - bowled via his elbow - the seamer beat the defences of Henry Nicholls before taking three late wickets to end the day with figures of five for 33.
Having been reduced to 3-23 in their reply, stand-in captain Ross Taylor and Luke Ronchi began a fightback before the latter was trapped in front for 35 as the rain began to fall.
Following a lengthy delay, Taylor edged Kumar to slip on 36 before Mitchell Santner and Matt Henry were beaten by balls that kept low in successive deliveries to give the right-armer his fourth five-wicket haul in Tests.
Jeetan Patel, on his comeback to Test cricket, and BJ Watling survived a nervy couple of overs in the gloom before the umpires called an early end to play.
India had earlier added 77 runs to their overnight score with Saha leading the way.
The wicketkeeper-batsman was involved in a 41-run eighth-wicket stand with Jadeja before the left-hander pulled Neil Wagner straight to Henry at fine leg.
Kumar was trapped in front by Santner for just five but Saha and Shami put on 35 for the final wicket before Trent Boult wrapped the innings up with the help of a fine catch by Henry.
