The England striker expertly rounded Edwin van der Sar on 66 minutes to drag the home side back into a game which the visitors looked to have in their grasp after a superb first-half display.
Steed Malbranque had fired the Londoners into a 15th-minute lead to reward their early dominance.
But the Wearsiders eventually staged a rousing comeback, and it was they who looked the more likely winners as time ran down.
Jean Tigana may feel his side deserved more than the one point, but Reid was a relieved man after seeing a fightback which reinforced his belief in his players.
It could have been so different as Malbranque in particular ran riot before the break, leaving the Wearsiders in his wake for long periods.
Reid went into the game with the result of a poll conducted by a fans' website professing little confidence in him, but the manager remained defiant.
He brought veteran striker Niall Quinn back to partner Phillips in attack and handed the experienced Stefan Schwarz a recall in midfield.
Tigana recalled goalkeeper van der Saar, John Collins, Sylvain Legwinski and Steve Marlet - with Maik Taylor, Sean Davis, Luis Boa Morte and Louis Saha making way.
Reid knew only an improved display from his players would silence his critics, but he got nothing of the sort before the break as Fulham took the game by the scruff of the neck.
The outstanding Malbranque and his midfield partner John Collins ran the opening 45 minutes - and, with Sunderland creating little where it mattered, the visitors could have wrapped up the points with barely half-an-hour gone.
Barry Hayles should really have put the Londoners in front with less than two minutes gone after he outstripped the home defence to collect Steve Finnan's pass - but he could not beat Thomas Sorensen.
His side had to wait just 13 minutes more for the opening goal, Malbranque skipping away from Schwarz, Bernt Haas and Stanislav Varga before driving a shot across Sorensen and into the top corner.
The Dane was grateful for the assistance of the crossbar on 34 minutes when Zat Knight rose above the crowd to get his head to Malbranque's corner.
Sunderland's failure to find the back of the net often enough to mount a serious challenge in this season's Barclaycard Premiership refused to go away,
Gavin McCann and Jason McAteer both squandering golden opportunities when through on goal to leave Reid scratching is head. But whatever he said at half-time certainly had the desired effect as his side upped their tempo and, little by little, forced their way into the game.
Quinn was unfortunate to see a well-struck volley cannon into Finnan, and Phillips warmed van der Sar's hands with a powerful volley after Knight's back-pass hit Alain Goma and fell into his path.
Knight chanced his arm from all of 40 yards - and was not too far off target - as Fulham looked to extend their lead, but it was Sunderland who were establishing a momentum.
Gavin McCann had a long-range shot blocked by Goma after Quinn had flicked on Haas' long throw - and Phillips sliced a speculative effort wide.
But the home side were starting to believe once again.
The equaliser eventually came with 66 minutes gone when Schwarz's pass allowed Phillips to race clear and round van der Sar with before slotting into the empty net.
From that point on, there looked to be only one winner - Phillips firing inches wide two minutes later and then hammering a volley at point-blank range into Legwinski.
As the Wearsiders committed men to attack in the hunt for victory, Fulham used their pace on the break to threaten. But Marlet wasted a glorious chance nine minutes from time when he went clear on the left and cut inside only to fire high over the bar.
In an eventful end to the game, McCann left the field with blood pouring from his nose after a clash with Knight and Fulham substitute Boa Morte was carried off just 13 minutes after replacing Hayles.