India resumed at its overnight 112-4, chasing an unlikely 445 for victory and needing its remaining batsmen to put together big scores to bat out the day, but the visitors lost early wickets and the cause was quickly lost.
James Anderson removed Rohit Sharma (6) with his third ball of the day before India captain Mahendra Singh Dhoni (6) edged an out-swinging delivery. Both catches were taken by Jos Buttler.
Anderson's double-strike saw him become the leading wicket taker in the series, with 16. He overtook India's Bhuvneshwar Kumar (15).
Moeen was then introduced in place of Stuart Broad to the bowling attack, and mercilessly cleaned up the rest of India's batting order.
He bowled out Ravindra Jadeja (15), before having Kumar caught by Anderson in the same over for a duck. He then bowled Mohammed Shami, also for a duck.
Ajiknya Rahane made his 50 with four off Chris Woakes, and finished unbeaten on a defiant 54, before Pankaj Singh (9) was bowled by Moeen, who finished with figures of 6-67.
"His level of improvement as a bowler, I haven't seen in such a short space of time," Cook said of Moeen. "To adjust from bowling county cricket to a holding role at international cricket with the lengths you have to bowl, the lines, the speed, he has learnt really well. When he got on a wicket that spun a bit, he proved he is an attacking option as well."
India captain Dhoni was frustrated that a team usually so adept at playing spinners was so poor against Moeen.
"We let Moeen bowl his line and length," Dhoni said. "There was a considerable amount of wear and tear which meant if you keep playing him, one odd ball will turn and with so many catching fielders around, you will lose your wicket."
Anderson's attention now turns to Saturday's disciplinary hearing for a physical confrontation with Jadeja during the first test.
If convicted of pushing and abusing Jadeja, he could be banned for four test matches. Anderson admitted he has tried to forget about it while playing for England since.
"The ECB (England and Wales Cricket Board) have done a great job in trying to keep everything separate," Anderson said. "When we get to the ground, it is all about the cricket."
Anderson said he has "no idea" if he will be playing the fourth test at Old Trafford, starting on August 8.
- AP