The home team was on the verge of overtaking their rivals in the sixth-place position after Harry Wilson scored his fifth goal of the season. However, Ouattara's skillful finish over Bernd Leno saved the day for both teams, as they maintained their unbeaten streaks that now stand at seven games each.
Raul Jimenez's header from Andreas Pereira's corner (40) had been cancelled out by Evanilson's brilliant turn and finish six minutes after half-time (51). Silva's men would once more demonstrate their battling qualities this term as Wilson ghosted into the Bournemouth box to restore the home side's lead (72).
But Bournemouth have developed a habit of scoring vital late goals through substitutes as replacement Ouattara would pen the final chapter of this helter-skelter contest on the banks of the River Thames.
The result means Fulham stay in eighth position while Bournemouth remain above them in sixth after 19 games this season.
TrendingMarco Silva cut a frustrated figure on the touchline at the final whistle, having witnessed his side drop more points from a winning position. It is 19 now, comfortably more than any other side and more than Fulham have let slip in each of the past two seasons.
And yet, they extended their unbeaten run to seven games, the first time the Whites have achieved that feat since under Mark Hughes over 14 years ago.
Silva wrote in his programme notes about finding solutions to absorb the absence of key personnel, but with Emile Smith Rowe named on the bench, those problems are now easing - and on Sunday, his players appeared to have found a route to prise the scalp of Bournemouth.
Wilson was a livewire throughout and it was a surprise when the Welshman was replaced late on with Fulham in the ascendancy. He took his goal very well, darted onto Robinson's cross to direct the ball in much the same way Jimenez had done in the first opener for the hosts' opener.
Backing up a first victory at Chelsea in 45 years with another success here was always likely to prove a difficult ask of the same players, and Iraola again found joy from his bench.
The visitors were the better of the two sides in the opening 45 minutes as Bernd Leno raced from his line to deny Antoine Semenyo. From the ensuing corner, James Hill was denied his first Premier League goal as referee Robert Jones spotted a foul by Semenyo on Joachim Andersen. It was soft.
Fulham made the most of their repreive and Bournemouth were rattled and could have been reduced to 10 men when Ryan Christie was only shown a yellow card for a reckless challenge on Antonee Robinson. VAR upheld the on-field decision.
Evanilson's goal was similar to the first he scored for Bournemouth earlier this season against Southampton, running off the shoulder of Andersen to almost break the net, and what impressed most about this latest fightback was how Fulham looked more likely to score the third when Ouattara's leveller arrived.
It came from a rare mistake from Leno as his slack clearance was picked up by Enes Unal. Semenyo cut inside and his attempted shot squirted free from Ouattara to calmly lift his finish into the net.