Though he scored a hat-trick against Shakhtar Donetsk last month, Jesus has scored only one league goal for City in a largely frustrating season to date.
Bernardo Silva's 16th-minute goal was City's only shot on target in the first half and was cancelled out by Bournemouth's Callum Wilson on the stroke of halftime.
Europe is a top priority for the club and Gundogan said the defeats to Jurgen Klopp's side last season showed how they can not afford any slip-ups if they are to win the Champions League.
And in his first full season, the highly-rated forward scored 17 goals in 42 games across all competitions as Pep found a rotational policy with Aguero that worked with aplomb as they romped to the Premier League title.
But, the Citizen coach is not anxious about that and asks him to relax and keep helping Manchester City to win. After the Champions League it is always tough.
Guardiola said afterwards to BBC Sport, "It was a good win". I know when I did it I was not happy to have done it.
Man City extend lead at top, Man United salvage draw | Bangkok Post: news
Sterling netted from close-range after Begovic spilled Danilo's shot, making it eight goals in six league appearances against Bournemouth for the City winger.
Jamie Vardy's 12th-minute penalty, after the striker himself was brought down, gave Leicester the lead.
Leicester climbed above visitors Watford into seventh after a 2-0 win at the King Power Stadium.
Newcastle's run of three straight wins came to an end after a 3-0 loss to West Ham at St James' Park.
City have lost just twice in the Premier League since the start of the 2017-18 season, and have already played away at three of their top-six rivals this year, with a trip to Chelsea coming on Saturday.
However, the hosts were reduced to 10 men after half an hour when Steve Mounie was sent off for a high challenge and Brighton took full advantage with goals by Shane Duffy and Florin Andone.More news: Neil deGrasse Tyson investigated by Fox, NatGeo Networks