Kohli revealed that Axar's heroics left Ishant Sharma and Jasprit Bumrah hilariously complaining about the lack of bowling opportunities in the third Test.
Axar was also adjudged Man of the Match for his brilliant display at the Motera strip.
The last time India won inside two days in a Test was in 2018, beating Afghanistan in Bengaluru.
Patel picked five wickets in the second essay, adding to his outstanding haul of six in the first innings as England collapsed at dinner break to record one of their worst batting shows.
Earlier in the first innings, Axar, sticking with a stump-to-stump line, had taken six wickets to blow away England for 112 runs.
In the last 100 years, no spinner has ever taken a wicket off the first ball of an innings. It has now happened twice in this very series.
Patel (21.4-6-38-6) grabbed his second five-for in his second Test while Ashwin (16-6-26-3) took three.
Indian spinners shot England out for just 112 inside two sessions with Ravichandran Ashwin and Axar Patel sharing the spoils after pacer Ishant Sharma opened the floodgates in his milestone 100th match
Axar's 6/38 stands as the second-best bowling figure by a spinner in Day-Night Tests after Devendra Bishoo's 8 for 49 against Pakistan in Dubai in 2016.
Ishant Sharma drew first blood after England opted to bat first in Ahmedabad before Axar Patel and R Ashwin picked three wickets between themselves as visitors went 4 down at Tea. Yet, Zak Crawley stood out with his 53-run knock.
Coming round the wicket, Axar bowled a slider that trapped Crawley in front on Day 1 of third Test at the Narendra Modi Stadium.
With the revelation of Axar, while it adds to India's bowling depth and ability to look beyond the famed pair of Ashwin and Jadeja, at present, it definitely creates a happy headache in opting for a spin combination irrespective of the format.
Axar is also the second Indian left-arm spinner after Dilip Joshi, who took 6 for 103 against Australia in Chennai in 1979, to join the list.
Axar Patel feels that critics of spinner-friendly tracks need to change their mindset as Indian cricket team has never complained about tackling seamer-friendly green tops overseas.
Spinner Axar Patel will make his Test debut in the second game of the four-match series against England in Chennai.
The BCCI on Thursday also shared a clip of Axar bowling in the nets, increasing his chances to feature in the Playing XI of the second Test.
Led by James Anderson's reverse swing sorcery, England bundled out India on just 192 runs in the second innings of first Test to go 1-0 up in the four-match series.
A repeat of the historic 2-1 Test series win in Australia would take India into the final of the World Test Championship.
Indian captain Virat Kohli didn't hide his disappointment when it came to Nadeem's performance during the post-match presentation.
Spinner Axar Patel has been ruled out of the first Test between India and England with a knee problem.
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