Raul Jimenez's sixth career goal against the Irons would ostensibly prove sufficient for Marco Silva's side, only for Ings to deny the hosts all of the London derby spoils with virtually the last kick of the game.
Separated by just one place in the bottom half of the Premier League table, Fulham had the look of a side with top-half aspirations in mind in the first half, but the same could not be said of their capital counterparts.
The Irons were being run ragged by Adama Traore, Alex Iwobi and Emile Smith Rowe in the opening exchanges, and the latter made a telling contribution midway through the first 45, teeing up Jimenez for his first of the Premier League season.
An off-colour West Ham had no shots on target in the first half, but the injection of pace provided by half-time substitute Crysencio Summerville saw the Irons belatedly turn the screw after the restart.
Julen Lopetegui's side still looked set to walk away from Craven Cottage empty-handed, only for an unexpected hero to emerge in the shape of Ings, who remarkably rescued a share of the spoils deep into injury time.
Both still languishing in the bottom half as a result of Saturday's stalemate, 11th-ranked Fulham are three places and one point better off than their visitors, who are yet to set the Premier League alight since Lopetegui took charge.
FULHAM VS. WEST HAM HIGHLIGHTS
Raul Jimenez goal vs. West Ham (24th min, Fulham 1-0 West Ham)
Fulham's dominance is rewarded, as Jimenez scores against West Ham United for the sixth time in his career.
Antonee Robinson intercepts an Irons pass on the left, and Andreas Pereira pings a pass over the top for Emile Smith Rowe, who just about manages to prevent the ball from going out of play on the left.
The Englishman nutmegs ex-Arsenal teammate Dinos Mavropanos as he picks out Jimenez, who gets across Max Kilman with equal ease to tap home from close range.
Danny Ings goal vs. Fulham (90 +5 min, Fulham 1-1 West Ham)
West Ham's second-half endeavour is finally rewarded with seconds remaining! In the fifth of five added-on minutes, Jarrod Bowen sends in an inviting ball into the Fulham box, which substitute Ings meets to dramatically secure a point for the Irons!
MAN OF THE MATCH - ANTONEE ROBINSON
Harshly denied a victory right at the death, Fulham left-back Antonee Robinson played a crucial role in Jimenez's first-half effort, sticking a foot in to stop a West Ham attack and triggering the move that led to the Cottagers' opener.
The American also put in six tackles and three clearances on the left-hand side, while winning both of his aerial duels and completing two successful dribbles in an all-around competent performance.
FULHAM VS. WEST HAM MATCH STATS
Possession: Fulham 54%-46% West Ham
Shots: Fulham 21-11 West Ham
Shots on target: Fulham 5-3 West Ham
Corners: Fulham 3-2 West Ham
Fouls: Fulham 15-18 West Ham
WHAT NEXT?
Fulham only have two full days to recuperate before travelling to Championship side Preston North End in Tuesday's third-round EFL Cup tie, and their next Premier League game comes at home to Newcastle United on September 21.
West Ham do not return to EFL Cup action for another week and a half, though, allowing Lopetegui's men to fully focus on next Saturday's lunchtime clash against Chelsea in their second successive North London derby.