Former Albion captain Robson, who was sacked as manager at The Hawthorns in September 2006, has faced calls for his head from United fans and is due to meet the club's plc chairman Kevin McCabe for a crucial meeting in Brussels on Wednesday.
But he can head into those talks boosted by a battling display against Albion, who failed to score in a home game for only the second time this season.
Albion, who have now won just one of their last five league games, have hit an early case of the promotion jitters.
They had plenty of chances to beat United and strengthen their promotion push but their free-scoring attack had an off night.
Robson's side were also lacking in front of goal as they had enough opportunities to have secured what would have been only their second league win in 10 games.
United started and ended the first half brightly, but in between were fortunate that the woodwork kept them on level terms.
Albion, who have made a habit of conceding early goals of late, were almost caught out again after just three minutes when Chris Morgan's close-range drive was blocked and David Cotterill sent the rebound wide at the far post.
United then had three major let-offs in the space of three minutes.
Chris Brunt whipped in a wicked left-wing cross in the fifth minute that Filipe Teixeira crashed against the foot of the post, before Jonathan Greening just failed to turn home the loose ball.
Then Teixeira turned provider with a left-wing corner that Martin Albrechtsen headed narrowly wide.
Having seen their early pressure come to nothing, Albion struggled to maintain their hold on the game and could easily have found themselves trailing at the break.
Luton Shelton threatened to embarrass Albion goalkeeper Dean Kiely from long range on the half-hour, before Matthew Kilgallon missed a golden chance in stoppage time.
Kilgallon was in acres of space when he met a Lee Martin corner only to steer his header narrowly wide.
Albion laid siege to the United goal at the start of the second half, but Albrechtsen saw his 59th minute dive blocked on the line by Stephen Quinn.
Quinn should then have set the seal on an impressive display when he was presented with a gilt-edged chance following a pinpoint left-wing centre by Gary Naysmith.
Quinn was unmarked but lazily lifted his shot over the bar from 16 yards.
It was really the last time that United threatened as Albion mounted a late rally but an out of sorts Kevin Phillips couldn't keep his 71st minute header following a Brunt cross.
Phillips then exchanged passes with Craig Beattie and only had Paddy Kenny to beat, but the Republic of Ireland international keeper comfortably denied the veteran striker.