The substitute wicketkeeper was part of three dismissals in New Zealand's first innings.
Bharat had to perform the role after designated wicketkeeper Saha was taken out due to the stiffness in his neck.
The substitute glovesman was involved in three dismissals that included the stumping of New Zealand's top-scorer Tom Latham (95) off Akshar Patel's bowling. He also grabbed the catches of Will Young (89) and Ross Taylor off the bowling of R Ashwin and Patel, respectively.
"I still remember Rahul Dravid raving about KS Bharat’s wicketkeeping skills. He told me that Bharat has good keeping skills, next only to Wriddhiman Saha in Indian cricket.
"It’s good to see him repaying the faith the selectors and the head coach. I think he has justified the confidence showed in him by the selectors and the coach," Laxman said on Star Sports after the day's play.
"In these quality spin conditions, if you don't have a reliable wicketkeeper then you're going to miss a lot of chances. What we saw today was great technique and great presence of mind, he didn't panic at all for someone who has come into this team recently and got to play only because Saha was injured.
"Fantastic experience for Bharat, this will give him a lot of confidence going forward in his career," Laxman added.
Coming to the match, India bowled out New Zealand for 296 to take a handy first-innings lead of 49 runs.
In the second innings, the hosts scored 14/1 before the stumps on Day 3. Mayank Agarwal (4*) and Cheteshwar Pujara (9*) were unbeaten in the middle as the home side extended their lead to 63 runs.