Paulo Dybala picked out Lukaku with a clever cross as the Belgium striker headed Roma in front in the fifth minute at the Stadio Olimpico.
Jack Grealish ensured Manchester City ended their four-game winless run as the Premier League champions survived a scare from Luton in their 2-1 victory on Sunday.
Manchester United were humiliated 3-0 at home by Bournemouth in the Premier League on Saturday as the pressure ratcheted up once again on manager Erik Ten Hag.
Former England goalkeeper Shilton is acknowledged as the record holder for the most competitive men's professional games, though the total is disputed. Some statistics websites say the tally is 1,390 but Shilton himself, on his X feed, gives it as 1,387.