Cook looked in good touch
Mohammed Shami (1/62) got Sam Robson's wicket but was the most disappointing of the pace trio erring in length on a number of occasions.
Ballance on his part hit 15 boundaries in his 204 ball unbeaten knock.
When the post-tea session began, Cook (82 batting) needed another few runs to score his first Test century in more than a year.
He began well, reaching the nervous 90s quickly, but once there the Indians mounted the pressure on him and it resulted in his dismissal.
Cook faced 231 balls and struck nine fours during his knock. It was his highest score since the 130 runs he made versus New Zealand in May 2013 and in doing so, reached 8257 runs in his career to overtake David Gower (8231 runs) and Kevin Pietersen (8181 runs) in the overall Test runs scored by Englishman.
Re: Cook looked in good touch
Cook gets his first half-century off the series
The English skipper first got the monkey off his back and got to his 36th half-century in the 31st over of the innings, two overs after the break.
He got to the milestone in 98 balls, with five fours, and was given a standing ovation by the Sunday crowd. Three overs later, the 100-run mark came up for the hosts.
In the 33rd over, bowled by Shami, there was a loud appeal against Cook for caught behind but the umpire didn't budge, even though replays later showed that there might have been a little edge in there.
Dhoni tried regrouping his team at the drinks-break with a pep-talk over a huddle, but it didn't really work out as Cook made his way through to his highest Test score in 13 innings.
The runs started coming easily as Ballance too reached his fifty in the 56th over, off 107 balls with 5 fours, two overs after the 100-run partnership had been completed between the two.
Jadeja dropped Cook on 15
In the morning session, Cook overcame his poor run of form thanks to a reprieve from Jadeja at third slip. The big moment for Cook came in the 12th over then, bowled by Pankaj, only for the all-rounder to spill an easy chance. The score read 25/0 at that time with Cook batting on 15 as Pankaj missed out on a maiden Test wicket.
To his credit, Cook then tightened up further and didn't give another chance to the Indian bowlers, who also improved their lines to the batsmen. The 50-partnership for the opening pair came up in the 19th over, when Cook played an immaculate cut shot for four. This was only his second four over nearly 90 minutes of batting, and 50 balls after his first boundary, showing how hard he was trying in the middle.
Shami removes Robson
But then India did break through as Shami returned in the 21st over to snap up Robson's wicket.
He edged a marginal out-swinger from back of length to third slip where Jadeja redeemed himself this time around. Robson scored 26 runs off 59 balls with 4 fours and added 55 runs for the first wicket.
India lead the five-match series 1-0 after winning the second Test at Lord's by 95 runs. The first Test in Nottingham ended in a draw.