Everton 0-2 Manchester United: Anthony Martial and Jesse Lingard help United secure a critical win.
Manchester United moved back into second place in the Premier League behind Manchester City after victory over Everton at a Goodison Park.
After a sequence of three draws, Manchester United boss Jose Mourinho was keen to end it and they were able to achieve it.
United clearly raised the tempo after an even first half, and Juan Mata had just hit the post with another swerving effort., United open the scoring with Martial curling from edge of the area in the 57th minute.
The United team was missing Lukaku, ruled out of a return to Everton after suffering a heavy knock to the head against Southampton, and Ibrahimovic, who has a knee injury, but they were still far too strong for Everton.
Lingard scored in similar fashion nine minutes to stoppage time.
Paul Pogba provided the assist for both goals, and was displayed man-of-the-match.
Everton manager Allardyce's problems are the same ones Ronald Koeman had before he was sacked in October - a horribly imbalanced squad with weaknesses exposed by the inexplicable failure to replace leading scorer Lukaku after his move to United.
Everton have ground out grim goalless draws against Chelsea and West Brom while they were spared defeat in the Merseyside derby at Liverpool by Rooney's late penalty.
Niasse is willing but very limited while Dominic Calvert-Lewin is in need of a rest after taking on such a heavy workload this season.
Everton face Liverpool at Anfield in the FA Cup on Friday at 19:55 GMT, before returning to Premier League action against Tottenham at Wembley on 13 January at 17:30.
United host Derby County in the cup on Friday, kicking off at 20:00. Their next league outing is on 15 January at home to Stoke at 20:00.