The substitute struck within five minutes of coming off the bench to rescue a 2-2 draw.
Kyle Bartley and Matheus Pereira had turned the game in the second half after the Baggies looked doomed following a wretched start.
Bobby Decordova-Reid's opener helped the Cottagers dominate before the break with the striker also hitting the post.
But Scott Parker's side are now winless in 10 Premier League games and sit four points from safety, with Albion a further two points behind.
A draw did little for either side's survival hopes and Baggies boss Sam Allardyce had underlined the need for four points from the next two games - with Albion going to rock-bottom Sheffield United on Tuesday.
He had locked them in the dressing room for a post-match inquest following the midweek thumping by Manchester City and the hosts needed an instant response.
Yet they suffered a nightmare start and were behind after 10 minutes.
Ademola Lookman's searching pass found Aleksandar Mitrovic and he held the ball up before slipping the unmarked Decordova-Reid in to lash past Sam Johnstone.
Two minutes later Johnstone saved from Ruben Loftus-Cheek as the flustered Baggies threatened to sink without a trace.
Fulham had not won since November but sliced Albion open at will and it should have been 2-0 after 19 minutes when Mitrovic and Decordova-Reid combined again.
Mitrovic's simple pass split the Baggies defence for Decordova-Reid to leave Kieran Gibbs floundering and hit the post with just Johnstone to beat.
Fulham's dominance forced Allardyce's hand and he replaced defender Dara O'Shea with striker Karlan Grant after just 24 minutes, although it made little instant impact.
Loftus-Cheek shot over and only Semi Ajayi's late intervention put Lookman off and allowed Johnstone to save when the Baggies were left wide open again.
There was no cohesion or steel to the hosts, with Fulham often allowed the freedom of The Hawthorns.
Sloppy Albion were lucky to only be a goal behind and Decordova-Reid skied over from the edge of the box as the Cottagers continued to create chances.
Fulham fully deserved their lead - a weak Grant effort the Baggies' only shot in the first half - but Albion levelled out of the blue two minutes after the break.
Pereira and Grant combined and Pereira's cross clipped Ola Aina for Bartley to steer in a first-time finish from six yards - the first goal at home under Allardyce to end a barren run of 466 minutes at The Hawthorns.
Albion had re-emerged refreshed - with intensity as they pressed Fulham back - and Pereira dragged wide from 25 yards.
It was a total contrast to the first half and the Baggies went ahead in the 66th minute.
Mbaye Diagne had been thrown on at the break after completing his loan move from Galatasaray on Friday and offered much-needed presence up front.
And the striker turned supplier when he went free down the right and Pereira reached his low cross before Aina to turn past Alphonse Areola from six yards.
Yet the lead only lasted 11 minutes when Harrison Reed bent in a brilliant cross and fellow substitute Cavaleiro's diving header levelled.
Johnstone gathered Mitrovic's header soon after but neither side found a late winner.
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