England international Marcus Rashford, only on the field for 10 minutes as a substitute, was sent off after 70 minutes for a headbutt on Phil Bardsley who had clipped the United man’s ankles in cynical fashion.
Just two minutes earlier, Mourinho had also seen Paul Pogba miss a penalty with the chance to put the visitors out of sight.
But Lukaku’s first-half double was enough for a well-earned three points as United bounced back from consecutive defeats to Brighton and Tottenham that had put Mourinho under the spotlight.
Lukaku opened the scoring on 26 minutes, after a period of bright and direct football from Mourinho’s team.
Alexis Sanchez was recalled to Mourinho’s starting line-up and responded by setting up the opener as Lukaku rose impressively between two defenders to head the Chilean’s cross past Joe Hart.
A second goal before the break looked inevitable, although it took until the final minute for it to arrive as Luke Shaw chased onto Sanchez’s back heel and into the Burnley area.
The ball eventually landed with Jesse Lingard on the edge of the area and, after his shot took deflections off both Charlie Taylor and Ashley Westwood, Lukaku was well positioned to turn and sweep in from six yards.
– Mourinho backed –
It was a rewarding first period for Mourinho and his team, not to mention the 2,400 visiting supporters at Turf Moor who showed their backing for Mourinho.
In contrast, there was a demonstration by fans against the club’s executive vice-chairman Ed Woodward with a plane flown over the Lancashire ground towing a banner that read: “Ed Woodward, a specialist in failure.”
It was a poignant comment from supporters who clearly blame Woodward’s lack of action in the transfer market for the club’s poor start to the season – rather than their high-profile manager.
That lacklustre opening to the campaign saw United in danger of losing three of their first four league games for the first time since 1986.
Mourinho, though, had looked relaxed even before kick-off, joking with officials in the tunnel and shaking hands with players from both teams as they exited the dressing rooms.
And within minutes of kick-off, that confidence looked well placed as United dominated.
Lingard, alone, was presented with three decent chances inside the opening nine minutes, one of which drew a slightly panicked save from Hart.
Hart also saved well from Lukaku, as he and Sanchez looked to convert a three-on-two break, while his opposite number David De Gea remained a virtual spectator.
The Spanish keeper was finally forced into action after 63 minutes, comfortably stopping Chris Wood’s header at the foot of his post.
But that was a rare respite for Burnley who needed Hart to save from a Sanchez half-volley and to parry a fierce shot from Shaw before the late controversies.
Lukaku should have claimed a hat-trick, being denied again by Hart, when clean through on goal, then actually rounding the keeper only to miss an open net under pressure from Ben Mee.