De Bruyne kicked things off with the opening goal and Iheanacho doubled their lead just before half-time.
Sterling added a third three minutes into the second half and Gundogan completed the scoring with the fourth goal with more than 20 minutes still to play.
The sending-off of Nolito in the closing moments, apparently for a head-butt, took some gloss off the celebrations but Guardiola was happy with the result, if not the performance.
"Our high pressing was not perfect today, we had problems with controlling the Bournemouth build-up, but we created a lot of counter attacks," he said.
"We are so lucky that we have Kevin de Bruyne with us. It was a fantastic performance (from him)."
Gareth Barry celebrated his 600th Premier League appearance with his side's opening goal at Everton after Middlesbrough had taken the lead through an own goal by goalkeeper Maarten Stekelenburg on 21 minutes.
Seamus Coleman put Everton in front three minutes before the break and Romelu Lukaku netted the third on the stroke of half-time.
Arsenal joined Chelsea and Liverpool on 10 points with a convincing 4-1 win at Hull, where a red card for Jake Livermore on 40 minutes ended the home side's hopes.
Sanchez missed the resultant penalty but Theo Walcott doubled their lead on 55 minutes and though Ryan Snodgrass pulled one back, Sanchez and Granit Xhaka, with a stunning debut strike from 30 yards, completed the scoring.
"The result and performance was key," Walcott said. "It was a shame we couldn't keep the clean sheet but four goals away from home, you are always going to get the win.
"We have a lot of great characters in the dressing room. We have been in this situation before but the next step now is Tuesday's (Champions League) match."
Islam Slimani scored twice on his debut as champions Leicester picked up where they left off in the Champions League in midweek, clinching their second league win of the season with a 3-0 win over Burnley.
A Ben Mee own goal finished things off for Claudio Ranieri's side, who moved into the top half.
"We knew we had to come back to the Premier League and had to carry on from where we left off in the Champions League," Leicester midfielder Danny Drinkwater said.
"We have been away for the first time as a club and it was new for us. So it was a relief to come back and not suffer a European hangover. Getting three points was the most important thing."
In the day's other game, Nacer Chadli scored twice as West Brom beat West Ham 4-2.
Salomon Rondon and James McClean scored the others and though Michail Antonio and Manuel Lanzini pulled two goals back in quick succession, it was too little too late.
West Brom joined Leicester in the top half while West Ham remain just above the bottom three.
- AAP
