Starting the day with a deficit of 143 rus after India finished their first essay at 347, the West Indies ended the day with a lead of 81 runs but they only have four wickets remaining going into the final day after Harbhajan removed Edwards (110) and Marlon Samuels (0) in the same over.
With Indian captain Mahendra Singh Dhoni opting for the second new ball, Praveen Kumar removed the dangerous Carlton Baugh (10) in the last over of the day to get his second scalp.
It was the 161-run third wicket partnership between Edwards and Shivanarine Chanderpaul (73 batting) that made the Caribbean batting look a lot better than what it appeared till now.
The debutant didn't take any undue risk during his innings although he played a few pull shots whenever it was pitched short. He faced 195 balls hitting nine boundaries and a pulled six off part-time leggie Mukund.
Edwards should consider himself lucky that he was able to reach the three-figure mark as he set out for a single pushing the ball to mid-off.
As Edwards was sent back by Chanderpaul, Harbhajan's throw deflected off his boots and resultant ovethrow helped him become 13th West Indian batsman to score century on debut.
However it was Harbhajan who finally broke the partnership that lasted 56.1 overs when Edwards trying to cut edged it to Dhoni. After a couple if deliveries, Marlon Samuels was adjudged leg before by umpire Kettleborough when the delivery looked like missing off.
From a comfortable 201 for three, the Windies slumped to 201 for five in no time. Baugh's dismissal in the final over was also a blow for the home team as it now depends on Chanderpaul to delay the inevitable unless the Weather Gods play spoilsport.
Edwards admirably filled in the big shoes of experienced Ramnaresh Sarwan, dropped for this Test, as took it upon himself to resurrect the innings when the home team were reeling at 40 for three.
The debutant showed admirable focus and a tight technique, without ever missing out on possibility of runs.
He walked in to bat with scoreboard reading 8 for one, after opener Kieran Powell was out, and soon he found two more batsmen being back in the hut as Indians looked to make short work of the hosts.
Opener Adrian Barath (6) was caught at fourth slip off Praveen while Darren Bravo (14) was holed out at mid-off as West Indies raised spectre of yet another miserable batting display.
The visitors though hadn't reckoned with Edwards, wrongly adjudged caught in the first innings. The doggedness and resolve of Chanderpaul was another aspect that made life difficult for the visitors.
The duo batted a wee bit slowly during the initial overs but once the visitors flagged, they began hitting out of trouble.
The Indians after a point of time seemed having resigned themselves to the second new ball much before it was due as the pitch eased out.
Edwards appeared very good with his cover drives and pulls and scored at will in the final session.
He was the dominant of the two partners and raised his half century from 87 balls with five fours while Chanderpaul was still in his 20s.
Chanderpaul had a lucky break on 25, when the total was 116 for 3, with Rahul Dravid dropping him at first slip off Harbhajan in the final over before tea.
The final session showed little change in pattern but for final 30 minutes as West Indies first wiped out the deficit and then went for a handy lead.
Chanderpaul, who dabbed, pushed and nudged for his runs, finally freed his arms and first straight drove Praveen Kumar and then on-drove Harbhajan for boundaries.
Edwards enjoyed a 'life' when a Harbhajan delivery hit the back of his bat and went to slips where Dravid was taken aback by the chance and let it go.
Earlier, West Indian speedster Fidel Edwards picked up yet another five-wicket haul but India still built a lead of 143 runs over the hosts.
The day began 15 minutes late due to early showers and there was another brief interruption of 15 minutes but it still was an action-packed morning.
India were all out after 10-odd overs of play, having resumed at 308 for 6, with West Indies conceding only 39 further runs.
Edwards picked up 5 for 103 in a courageous effort of 28-odd overs in searing heat. This is his second successive five-wicket haul and he now has 18 wickets from the series.
The procession of batsmen began from the second ball of the morning itself when Harbhajan Singh misconnected a pull off Fidel Edwards and wicketkeeper Carlton Baugh strolled across on the onside for the simplest of chances.
The other overnight batsman, skipper Mahendra Singh Dhoni (74) and new man Praveen Kumar (23) then batted aggressively to add 31 runs from 47 balls for the eighth wicket.
Praveen was the more aggressive of the two, as he picked up two boundaries in one Edwards over, a thrust down the midwicket fence and a Chinese cut past his stumps, before he lashed a six off leg-spinner Devendra Bishoo.
Dhoni himself drove a shot over mid-off fielder as runs came at a fast clip. A brief rain interruption though disrupted the run-making pattern as Dhoni pulled an Edwards delivery into the hands of deep square leg fielder after resumption.
The skipper batted for 133 balls and hit four fours in the process.
Praveen's entertaining knock ended when he swished at a delivery from Bishoo and gave a catch to the point fielder.
Edwards then accounted for Ishant Sharma to bring a lid over the Indian innings.
Scoreboard at stumps on the fourth day of the third and final cricket Test between India and West Indies being played here.
Scoreboard
West Indies (Ist Innings) 204
India (Ist Innings) 347
West Indies (2nd Innings)
A Barath c Kohli b Kumar 6
K Powell c Raina b Sharma 4
K Edwards c Dhoni b Harbhajan 110
D Bravo c Kumar b Harbhajan Singh 14
S Chanderpaul batting 73
M Samuels lbw Harbhajan 0
C Baugh c Mukund b Praveen Kumar 10
D Sammy batting 1
Extras (b-5, nb-1) 6
Total (For 6 wkts; 83 overs) 224
Fall of wickets: 1-8, 2-10, 3-40, 4-201, 5-201, 6-223
Bowling: Praveen 15-5-29-2, Ishant 16-4-38-1, Munaf 17-3-58-0, Harbhajan 25-6-61-3, Raina 8-1-19-0, Mukund 2-0-14-0.