Root made 48 and nine in the defeat at Lord's, twice falling to loose shots.
He made a Test-best 254 at Old Trafford as England amassed 589-8 before reducing Pakistan to 57-4 on day two.
"They were poor dismissals. I've worked hard this week to take them out of my game. It was very satisfying to finally make one count," said Root, 25.
"To be able to bounce back straightaway and make a really big score, that's what you dream of," he told BBC's Test Match Special.
At Lord's Root top-edged a slog-sweep off Yasir Shah in the first innings and pulled Rahat Ali to deep square-leg in the second.
Root, unbeaten on 141 overnight, faced 406 balls and batted for more than 10 hours in Manchester to allow England to declare after tea.
Former England batsman Geoffrey Boycott described Root as "one of the best four or five players in the world - he'd get in any world XI".
Boycott added: "He was composed, orthodox, textbook - it was a very fine innings.
"A few days in politics is a long time. A few days in cricket can change everything. It hurts when you lose and it hurts when you play awful shots like at Lord's."
Root put on 103 with Chris Woakes (58) for the fifth wicket and 106 with Jonny Bairstow (58) for the seventh as England amassed their second highest total against Pakistan.
Woakes undermined Pakistan's reply with three wickets and Ben Stokes one in the 24 overs possible before the close to leave the tourists trailing by 532.
"We put a lot of pressure with the new ball [on Pakistan] and the guys after that took the rewards," said Root. "The way the guys bowled was fantastic.
"Pakistan will keep looking at the scoreboard. They've got a big mountain to climb."
Pakistan batting coach Grant Flower said: "There was weariness, and a huge score like that plays mind tricks.
"But that's what happens in Test cricket, so it's no excuse."
England batsman Joe Root said his failures in the first Test against Pakistan made him "hungrier", after he scored a career-best 254 in the second.