In reply to the hosts' 316, the Black Caps struggled to 128-7 when bad light brought play to a close, Bhuvneshwar taking the acclaim of the Kolkata crowd for his 5-33.
It was the first time the seamer had taken five wickets in a home Test, and his efforts put his side in a strong position to take an unassailable 2-0 lead in the three-match series, following their 197-run success in the opener.
India resumed on 239-7 following an impressive display with the ball from the Black Caps on day one, and they managed to add 77 runs more to edge beyond 300.
Without captain and star batsman Kane Williamson due to illness, New Zealand's top order struggled in their reply, falling to 23-3, before a stand of 62 between stand-in skipper Ross Taylor and Luke Ronchi helped steady the ship.
However, Ronchi was lbw to Ravindra Jadeja (1-17) for 35, a dismissal that immediately preceded a rain delay of over two hours and when play resumed Bhuvneshwar removed Taylor, Mitchell Santner and Matt Henry - the latter pair in successive balls.
Bhuvneshwar's late burst ended a fine day for India, who began with the need for lower-order runs. Jadeja was unable to make a telling contribution, hooking Neil Wagner (2-57) to long leg for 14.
Santner (1-83) pinned Bhuvneshwar in front for five soon afterwards and the innings was wrapped up when Henry took an excellent diving catch at fine leg to dismiss Mohammed Shami (14) off Trent Boult (2-46).
New Zealand's reply began poorly, Tom Latham falling lbw to Shami (1-46) for just a single, with Martin Guptill (13) playing on in the next over to give Bhuvneshwar his first victim.
Williamson's replacement Henry Nicholls departed for just one, also chopping Bhuvneshwar onto his stumps, and Ronchi and Taylor (36) fell either side of the lengthy rain break, the latter edging to slip.
Bhuvneshwar then trapped Santner lbw and bowled Henry, Jeetan Patel playing and missing at the hat-trick ball as India took firm control with the Black Caps still 188 in arrears.
Scores in the match entering day three are: India 316, New Zealand 128-7.