At the Etihad Stadium, City boss Pep Guardiola made seven changes, but Oleksandr Zinchenko got the scoring started against fourth-tier Vale after 20 minutes.
Port Vale striker Tom Pope had memorably mocked John Stones last year, claiming he would love to play against him after being underwhelmed by the City defender's performance during an England match.
And with Stones returning for City after a month out with a hamstring injury, the 34-year-old Pope backed up his social media taunts by equalising with a glancing header in the 35th minute.
Sergio Aguero restored order when he fired City back in front with his 15th goal of the season three minutes before half-time.
Harwood-Bellis sealed the win from Stones' flick in the 58th minute and Phil Foden got the fourth in the 76th minute.