With the rain bucketing down in the north-west neither side was able to dictate the tempo in an even contest, and it seemed the spoils would be shared when Lukaku's header came back off a post.
But a defensive lapse from the visitors allowed an unmarked Lukaku to flick on for substitute Martial, whose bobbled effort went beyond Hugo Lloris for the Frenchman's fourth goal as a substitute this season.
More importantly for United, Martial's strike maintained their 100 per cent winning record at home this season, in which they have not conceded either, as they moved three points clear of third-placed Spurs in the Premier League.
Rivals Manchester City retained a five-point cushion at the top of the standings but were given a brief scare before running out 3-2 winners at West Brom.
Free-scoring City went ahead in the 10th minute when Leroy Sane slammed home but they were almost immediately pegged back when Gareth Barry's lofted ball allowed Jay Rodriguez to outmuscle John Stones and lift over Ederson.
Normal service resumed in a frantic opening quarter of an hour as Fernandinho's deflected strike crept past Ben Foster, whose net was breached again in the 64th minute when Raheem Sterling slotted home from David Silva's low cross.
Matt Phillips profited from a Nicolas Otamendi error to score in stoppage time but City's eighth successive league win was never in doubt.
Sam Clucas netted his first goal since swapping Hull for Swansea in the summer, latching onto Tammy Abraham's through ball and slotting under Petr Cech and it was a lead they hung on to at half-time to the chagrin of the home support.
Arsenal were rejuvenated on the resumption and only needed five minutes to level matters as Sead Kolasinac fired across former Gunners goalkeeper Lukasz Fabianski, while Aaron Ramsey netted the winner from Kolasinac's cutback just before the hour.
Liverpool's second-half blitz brought Huddersfield crashing back down to earth following the Terriers' win over Manchester United last week.
Darren Fletcher grabbed his first goal for Stoke and his 16th-minute volley proved decisive in the Potters' surprise 1-0 victory at Watford, who dropped one place in the table to seventh following Liverpool's win.