There was an ill-tempered start to the second half after Rashford went down holding his face following minimal contact from the arm of Hegazi.
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Lukaku, who spent the 2012-13 season on loan at West Brom, didn't celebrate after heading in Marcus Rashford's inswinging cross in the 27th minute. But like an unwanted bubble of digestive gas, United's defensive frailties rose up from nowhere.
United made a dominant start with West Brom guilty of squandering cheap possession on a frequent basis.
Lukaku is back to scoring goals.
Lukaku is lurking in the area between the two West Brom centre backs, just as he was against Bournemouth. Is he not disappointed that he hasn't, yet? He scored against Bournemouth during the week, but it was only his second Premier League goal since September.
Mourinho told the media about the possible departure of some of his players in January, saying, "If there is an unhappy player in the team, and if he can secure the right offer, then I will not stand in his way and I will allow him to leave". He seemed pleased enough when Lingard doubled United's lead. It simply got worse (more boring) and the safest pair of hands in England had turned into a butterfingered mess by the time West Brom decided that their club shop could stand the drop in profits that would come from sacking Pulis.
Lingard's fourth goal in five games was a belated birthday present to himself after turning 25 on Friday. "We brought the ball into the opposition half".
Brunt delivered a corner from the left, United failed to clear and Barry profited from an ugly scramble to poke home the club's first goal under Alan Pardew and his first for Albion.
Mourinho already has his old teammates Chris Smalling and Phil Jones available to choose from, not to mention Axel Tuanzebe, Marcos Rojo and summer signing Victor Lindelof.
He had a similarly impressive strike rate last season with Everton: he was the Premier League's second top scorer behind Kane, but six players had more shots than him throughout the campaign.
The France international knows he needs game time to make the 2018 World Cup and has put West Ham and Crystal Palace on alert with his plans. The Gunners are without Shkodran Mustafi, who suffered a groin injury in their defeat to United at the Emirates recently, but the German is likely to be back for their game against Liverpool next weekend.