The Premier League champions continued their winning start to the new campaign after beating West Ham United in their opening league game last week.
It was a tough afternoon for Bournemouth who were unable to replicate their performance in last week’s victory over Aston Villa.
Erling Haaland, who was making his home debut, created the opening goal by brushing past two defenders to set up Gundogan after 19 minutes.
The hosts doubled their lead after 31 minutes when De Bruyne produced a fine finish with the outside of his right foot into the bottom corner.
The Belgium international then turned provider and set up Foden to score the third goal just before the break.
The second half turned out to be a quieter affair but City grabbed a fourth with 10 minutes remaining when Lerma turned Joao Cancelo’s cross into his own net.