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With this win, Jose Mourinho became the first manager to win a European trophy with four different clubs.
Roma confirmed the 40 million euro ($47.08 million) deal for the 23-year-old, who joins Jose Mourinho's team in search of game time.
The news comes the morning after the London club said it was one of 12 clubs forming a breakaway European Super League.
United edged their heavyweight tie with AC Milan as Paul Pogba came off the bench to score the goal that secured a 1-0 win in the San Siro for a 2-1 aggregate triumph.
The current roster is probably the most balanced line-up the club has ever seen.
Bookmakers William Hill reckon the Portuguese coach an 8/11 bet, with Southampton's Mark Hughes and Cardiff City's Neil Warnock the second-favourites to part ways with their employers at 7/1.
The 25-year-old - who scored in the 4-2 World Cup final win over Croatia - has agreed terms with Barcelona for a deal worth £89.5 million over five years, according to the Daily Mail.
South Korean Son has often been the go-to player when lead striker Harry Kane is injured and again he stepped up to convert in the 87th minute after being taken down by keeper Angus Gunn in a frenetic finale.
Son Heung-min and Harry Kane each scored twice for Spurs while United's Anthony Martial was sent off in the 28th minute as Jose Mourinho celebrated a stunning victory on his return to the club which sacked him 22 months ago.
Tottenham Hotspur, Antwerp, Lille, Villarreal, Braga, Dinamo Zagreb, Red Star Belgrade, PSV Eindhoven, Granada, Bayer Leverkusen, Slavia Prague, Rangers and Benfica joined Milan in booking their places in the last 32 on the night.
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