The Red Devils eased to a 3-1 victory over Bournemouth in their Premier League opener at Vitality Stadium last weekend after goals from Juan Mata, Wayne Rooney and Zlatan Ibrahimovic.
Southampton could only manage to draw 1-1 with Watford at St Mary’s as new manager Claude Puel took charge of his first Premier League game.
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The Saints have won at Old Trafford in the past two seasons under former manager Ronald Koeman, who took charge of Everton earlier this summer.
Former Liverpool defender Lawrenson doesn’t envisage the Saints making it three successive wins at the Theatre of Dreams on Friday night.
“Manchester United’s 3-1 win at Bournemouth was a good result away from home so early in the season, especially under a new manager and with new players in the side,” Lawrenson told BBC Sport.
“Southampton have won 1-0 on their last two trips to Old Trafford, but I do not see them making it three in a row.
“With Paul Pogba set to make his debut, this United team already feels different to when Louis van Gaal was manager and, at the moment, Saints are not the same side they were either.”
Jose Mourinho’s side are at the top of the Premier League table thanks to their 3-1 win against Eddie Howe’s Bournemouth side on the south coast.
The Red Devils haven’t won the Premier League title since 2013 when Sir Alex Ferguson secured his final top-flight title as United manager.
United are ahead of Chelsea and Liverpool in the Premier League table.
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