West Brom were 12 points adrift of safety, winless in the league since January prior to kickoff and under the watch of caretaker boss Darren Moore, but offered the hosts an early warning shot with a couple of chances.
Jake Livermore did not generate enough power behind his shot to trouble David de Gea, after he had checked back onto his left foot in the box, while Salomon Rondon headed over from James McClean's cross.
The hosts struggled to cut through Albion's compact defence and when they did found an equal in former goalkeeper Ben Foster, who blocked from Romelu Lukaku at point-blank range and kept out Alexis Sanchez's follow-up with his legs.
Lukaku would have had another opportunity at a one-on-one with Foster on 36 minutes had his heavy first touch not allowed the Albion stopper to race off his line before the Belgian could get a shot away.
The McClean-Rondon combination continued to prove effective for the Baggies and it created them a chance either side of half time, but De Gea positioned himself well to make a comfortable save on each occasion.
Jose Mourinho hooked Ander Herrera and Paul Pogba for Jesse Lingard and Anthony Martial before the hour mark and eventually United started to find a greater cutting edge. Lukaku's powerful and low 67th-minute header was denied by a diving save from Foster, while Lingard's 20-yard shot whistled past the post.
Just as the Red Devils were beginning to knock on the door, West Brom took a shock lead when Rodriguez converted a simple close-range header from a corner, having been inadvertently teed up by Nemanja Matic.
Fearing a slip-up that would crown their rivals champions, the hosts threw everything at Albion in the final 15 minutes but could not create a clear-cut chance to delay City's coronation as they did last week.
Chris Smalling was United's unlikely hero against the Citizens and came the closest to salvaging them a draw when he met Antonio Valencia's stoppage-time cross with a header which was saved low by Foster.
This scalp keeps Albion afloat for another week but relegation to the Championship still seems a certainty despite their best result of the season.
MAN UNITED (4-3-3): De Gea; Valencia, Smalling, Lindelof, Young (Rashford, 75'); Matic, Herrera (Lingard, 46'), Pogba (Martial, 58'); Mata, Sanchez, Lukaku
WEST BROM (4-4-1-1): Foster; Nyom, Dawson, Hegazi, Gibbs; Phillips (Krychowiak, 77'), Livermore, Brunt (Yacob, 90'), McClean; Rodriguez; Rondon (Sturridge, 85')