South Africa began the day at 36 for three but a rearguard action through Keshav Maharaj (72) and Vernon Philander (44) helped them cross the 250-run mark.
Ashwin's 350 in just 66 Tests places him fourth in the all-time list for India behind Kumble's 619, Kapil Dev's 434 and Harbhajan's 417.
While both had reached the milestone in their 66th Test, Ashwin took 18 innings more than Muralitharan, but bowled fewer deliveries than the former Sri Lankan cricketer
Ravichandran Ashwin took eight wickets in the first Test against South Africa and equalled the world record for the fastest bowler to take 350 wickets in Tests.
Ashwin also said that his maiden Test wicket will always hold a special place but there are a few others up there in the list.
Shami also hailed India captain Virat Kohli for giving the bowlers freedom to decide the length of their spells.
India defeated South Africa by 203 runs in the opening Test to take a 1-0 lead in the three-game series.
Indian spinner Ravichandran Ashwin has equalled bowling great Muttiah Muralitharan's feat during the fifth day of the ongoing Test against South Africa.
Ravichandran Ashwin is just one wicket short of equaling Muralitharan's world record of being the fastest to 350 Test wickets.
The 33-year-old, who has been playing only one format for India since July 2017, last played the Adelaide Test from December 6 to 10, the first of the four on the tour of Australia.
Dean Elgar and Quinton de Kock kept the skilful Indian spinners at bay with memorable hundreds, orchestrating South Africa's spirited counter-offensive against the fancied hosts in the series-opening Test.
The Proteas dominated the first two sessions of the day but the last session was clearly shared as the spin duo of Ashwin and Ravindra Jadeja clawed back.
According to the legendary Indian batsman, Ashwin should always be certainty whenever India play Test cricket
With India skipper Virat Kohli declaring the innings at 502/7, the idea was to give the spinners a go before stumps to assess what the pitch has on offer and the duo of Ashwin and Jadeja didn't disappoint.
Ashwin may not have been a regular in the Indian Test team during their overseas tours, but Sachin Tendulkar has insisted that the bowler remains an important part of the team.
Keshav Maharaj, who is on his maiden tour to India, has said that the upcoming three-match Test series against the World No.1 side will decide if he belongs to international cricket.
Ashwin led KXIP for a couple of seasons without much success and there has been news that the Preity Zinta co-owned franchise is ready to trade him off.
On his selection in the first Test, Shastri said that Jadeja's ability to bowl on a damp wicket in the first session of a Test match was factored in.
Rohit Sharma and Ravichandran Ashwin didn't find a place in the playing XI for the first Test against West Indies, and vice-captain Ajinkya Rahane revealed the reason behind the decision.
Ravichandran Ashwin is currently playing for Dindigul Dragons in the Tamil Nadu Premier League.
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