West Brom are now three points from safety with 11 points from 18 matches.
Two penalties from Matheus Pereira helped the Baggies earn a 3-2 victory at their biggest rivals Wolverhampton Wanderers and prompted hopes that Allardyce could lead one of the rescue acts that has seen him dubbed "Fireman Sam".
He headed home in the 51 minute to make it 2-2.
Wolves led 2-1 at halftime but a fine display in the second half allowed West Brom to score twice in four minutes to take all three points, giving Allardyce his first victory at the club and keeping his unbeaten record against Wolves intact.
Brighton notched their first English Premier League win in almost two months thanks to Neal Maupay's first-half goal in a 1-0 victory at Leeds on Saturday.
David Button replaced Johnstone in goal, while new signing Robert Snodgrass made his Albion debut in Phillips' absence.
Wolves fell behind in the eighth minute with a self-inflicted wound.
Michail Antonio scored in his first English Premier League start for West Ham since November in a 1-0 win over Burnley on Saturday.
That made it nine consecutive league games in which Wolves had fallen behind.
Sloppy defending was Albion's downfall again as Wolves went in front five minutes later.
The visitors sealed the win when Robinson won another penalty after drawing a foul from defender Conor Coady and Pereira scored from the spot again.
Neal Maupay netted after 17 minutes with a tap-in after Alexis Mac Allister and Leandro Trossard combined to open up the Leeds defence.
Later on Saturday, Chelsea travel to west London neighbours Fulham, while Leicester host Southampton.