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A late goal by substitute Ole Gunnar Solskjaer - his 100th for the club - spared United's blushes in what has been a tumultuous week for the Old Trafford giants.
West Brom were reduced to 10 men midway through the second half when Derek McInnes was sent off for a second bookable offence, but had performed valiantly until that point.
Megson's side can take heart from a spirited display.
However, questions will again be asked of United as to why it took them until the 78th minute to break down supposedly inferior opposition.
In the aftermath of United's Champions League debacle, Sir Alex Ferguson opted to fortify his midfield, giving Nicky Butt a place in the starting line-up.
Despite an anonymous display in midweek, Juan Sebastian Veron's services were retained and the Argentine shone in the opening exchanges.
His passing and vision were at times exquisite, complemented well by the combative Butt.
Ryan Giggs and David Beckham stayed wide, leaving captain Roy Keane as the most advanced United player other than Ruud van Nistelrooy.
And Keane, in the media spotlight once more following his admission that he deliberately injured Manchester City defender Alf Inge Haaland, came closest to breaking the first-half stalemate.
After some excellent interplay with Beckham, Butt set up Keane, but the Irishman failed to get enough purchase on his header and a good chance went begging.
Keane's second attempt, again a header, came courtesy of a Beckham cross and proved far more testing for keeper Russell Hoult, who gathered at the second attempt.
Giggs and Butt both wasted opportunities to open the scoring, while West Brom's �2m man Jason Roberts proved the only real threat to United's defence in a first half full of endeavour but lacking any real penetration.
After the interval, Roberts seized the initiative for West Brom, who were visibly buoyed by the 0-0 half-time scoreline.
However, the Baggies' early charge soon dissipated as United launched wave after wave of attacks in search of the opener.
Ferguson replaced Veron with striker Solskjaer and United reverted to their usual 4-4-2 formation.
But it was the introduction of Paul Scholes proved decisive for United. As soon as the England international took to the field, the hosts regained their usual swagger.
And a typical piece of Scholes skill in the West Brom area created an opening for Solskjaer, which the Norwegian striker netted with his customary aplomb.
It was a satisfactory afternoon's work for United, but much work remains to be done if the Reds are to regain their domestic dominance.
Lost 1-0 vs Zalaegerszegi (A)
Won 5-0 vs Aarhus (A)
Won 3-0 vs Parma (A)
Lost 1-2 vs Ajax (A)
Won 2-1 vs Valerenga (A)
Won 3-2 vs Bournemouth (A)