Jansen recorded figures of 6-48 in 19.5 overs, his fourth five-wicket haul in his 21st appearance, as the home team were skittled out for just 201, a deficit of 288 runs after the Proteas' first innings total of 489.
Proteas captain Temba Bavuma opted not to enforce the follow-on and the visitors reached 26 without loss at the close of play, an overall lead of 314 and two days of the Test still remaining.
India resumed the morning on nine without loss and opening pair Yashasvi Jaiswal and KL Rahul looked fairly comfortable as they extended their partnership to 65.
After an initial burst with his seamers, Bavuma turned to spin and it worked a treat as Keshav Maharaj had Rahul caught at slip for 22.
Jaiswal (58) was attacking and reached his fifty off 80 balls as the left-handed batter and number three Sai Sudharsan took the total up to 95.
It was then that off-spinner Simon Harrmer got a delivery to grip off the pitch and Jaiswal found an edge that looped to Jansen at point who completed a good catch diving forward.
The tourists then completed a fabulous first session as Harmer got rid of Sudharsan (15) and then Jansen had Dhruv Jurel caught at mid-off after the batter made a complete mess of playing a short ball.
The hosts went to the tea break on 102-4 and it did not get better after the interval as skipper Rishabh Pant and Nitish Kumar Reddy fell in quick session as a fired-up Jansen continued to make inroads.
The score became 122-7 soon after as Jansen got his fourth wicket with Ravindra Jadeja also failing to control a hostile short delivery from the left-arm paceman.
India were in a shambles but Washington Sundar and Kuldeep Yadav showed some reliance and took lunch on 174-7.
The home duo put on a defiant partnership of 72 but Harmer made the breakthrough early in the final session as Sundar walked back to the pavillion for 48.
With the last recognised batter out, Jansen returned to the attack and completed his fifer with the wicket of Kuldeep and hhad Jasprit Bumrah caught behind as the innings came to a close.
Aiden Markram and Ryan Rickelton then batted out eight overs without any major alarms to make it perfect day for the Proteas as they chase a first series win in India in 25 years.
India's Sanju Samson believes clarity in his role and position has given him the confidence to play the type of innings he did against England in the T20 World Cup semifinal.
In a high-octane thriller at the iconic Wankhede Stadium in Mumbai on Thursday, India held their nerve to defeat England by 7 runs in the second semi-final of the ICC Men's T20 World Cup.
The West Indies cricket team has been stranded in India for more than four days, prompting head coach Daren Sammy to express his frustration on social media on Thursday.
Proteas captain Aiden Markram cut a dejected figure as he fronted up to the press in the wake of their T20 World Cup semifinal loss to New Zealand on Wednesday.
