The England captain's position has come under much scrutiny in recent weeks, after a torrid run both personally and for the team, but his 231-ball innings helped the hosts reach 247-2 by the close at the Ageas Bowl in Hampshire on Sunday.
Gary Ballance once again showed his competency at Test level, compiling a chanceless century and finishing unbeaten on 104 alongside Ian Bell (16 not out).
England, winless in 10 Test matches since victory against Australia at Durham in the home Ashes series of 2013, have seen their recent problems exacerbated by Cook's dismal form.
Without a Test century since his 130 against New Zealand at Headingley in May last year, Cook appeared set to end a run of 27 innings without reaching three figures, until he got a thin edge down the leg side to an innocuous ball from Ravindra Jadeja (1-34 from 22 overs).
After winning the toss, it seemed as though Cook's luck had taken a turn for the better when he was dropped at third slip by Jadeja on 15.
That chance came off the bowling of India debutant Pankaj Singh (0-62 from 20), who had been called up for the injured Ishant Sharma.
India's other change saw batsman Rohit Sharma replace all-rounder Stuart Binny.
Jos Buttler, Chris Woakes and Chris Jordan came in to England's side at the expense of Matt Prior, who decided to step down for the rest of England's home season after the previous Test, Ben Stokes and Liam Plunkett.
After being given a life by Jadeja, Cook made steady progress with Sam Robson before the latter departed for 26, held in the slips by Jadeja off Mohammed Shami (1-62) to leave England 55-1.
Robson's demise brought Ballance to the crease, and the Yorkshire left-hander was largely untroubled as he shared 158 with his captain to put England firmly in control.
Although Cook was unable to make three figures, Ballance did so in the 84th over, guiding a Shami delivery down to third man.
In doing so, the Yorkshire batsman became the fifth England player to compile three Test centuries in his first six appearances.
He and Bell were subsequently able to reach the close unscathed, although the latter was perhaps fortunate to survive an appeal for lbw before he had scored.
England will be confident of posting a big first-innings score on Monday and putting the tourists, who lead the five-match series 1-0, under pressure.