Defender Stones, who had managed one goal in his first 170 Premier League appearances, has now netted four in his last 10.
His close-range effort broke Fulham’s resistance two minutes after the interval and after Gabriel Jesus capitalised on a Fulham mistake to make it 2-0, Aguero netted a penalty to grab his 181st City league goal, but first for 14 months.
While City have all but sewn up their third title in four seasons, the battle for a top-four spot continues to be unpredictable and looks set to go to the wire.
Chelsea’s 0-0 draw at Leeds United made it 12 games without defeat since Thomas Tuchel replaced Frank Lampard in charge.
But it opened the door for Everton later to close the gap on Chelsea who occupy fourth spot.
Victory against Burnley would have put Everton into fifth, two points behind Chelsea having played a game less, but Carlo Ancelotti’s side suffered another home stumble.
Chris Wood and Dwight McNeil gave Burnley a 2-0 lead and while Dominic Calvert-Lewin did pull a goal back for the hosts they could not avert a seventh league defeat at Goodison Park.
“It’s a big disappointment but we can’t give up. There are 10 games left and we fight until the end,” Ancelotti said.
Everton remain in sixth spot with 46 points from 28 games, two points behind West Ham United who could go level on points with Chelsea, and have a game in hand, on Sunday if they win at second-placed Manchester United.
Tottenham Hotspur, who have 45 points from 27 games, could go fifth on Sunday if they win at Arsenal.
Burnley’s win eased their relegation fears as they are now seven points ahead of third-from-bottom Fulham who had looked impressive in the first half against City before crumbling.
“Disappointing. I asked the team to be brave and show courage against the team who will be champions — and we were,” Fulham boss Scott Parker said. “There’s no denying the mistakes we gave away for the goals were sloppy.
“We will dust ourselves down and move on.”
Bottom club Sheffield United parted ways with manager Chris Wilder on Saturday and while they look doomed to the drop, West Bromwich Albion are also running out of time.
Their 1-0 defeat by Crystal Palace left them eight points behind Fulham and 17th-placed Brighton and Hove Albion.
“There’s 27 points available and 20 needed,” West Brom manager Sam Allardyce, who has never been in charge of a relegated team in the top flight, said.
Palace captain Luka Milivojevic decided the game with a VAR-awarded penalty while the other talking point was Palace’s Wilfried Zaha opting not to take a knee before kickoff — the first Premier League player not to make the anti-racism gesture this season.
“There is no right or wrong decision, but for me personally I feel kneeling has just become a part of the pre-match routine and at the moment it doesn’t matter whether we kneel or stand, some of us still continue to receive abuse,” he said.
“I now just want to focus on football and enjoy being back playing on the pitch. I will continue to stand tall.”