Gio Reyna and Mahmoud Dahoud had their own narrow misses from speculative long-range shots, while Zenit goalkeeper stopped an effort from Marco Reus.
The visitors took advantage as some Pulisic trickery allowed Werner to lay off to Ziyech, who tucked a neat finish beyond a crowd of defenders.
Solskjaer threw on Rashford, Fernandes and Scott McTominay within the space of a few minutes to change the dynamic and United reaped the rewards.
Messi, 33, set up Ousmane Dembele to put Barca in front after quarter of an hour in Turin, adding a second himself from the penalty spot in stoppage time.
Marseille, on the other hand, has not even scored yet and is already starting to stare elimination in the face.
Heading into the 87th minute against Borussia Mönchengladbach, Madrid was losing 2-0 and on course for four straight losses in the Champions League for the first time.
Morocco international Hakimi joined Inter from Real Madrid in the summer after a successful two-year loan at Borussia Dortmund.
Chelsea witnessed a goalless draw against Sevilla in the Champions League here on Wednesday.
If he plays like tonight, I'll have no complaints: Ronald Koeman satisfied with Lionel Messi's performance
Messi scored one goal during the match and he became the first player in the Champions League to score in 16 consecutive seasons.
The captain is nursing a left knee injury and did not practice with the squad on Tuesday.
Lionel Messi opened the scoring and two teenage teammates also chipped in to lead Barcelona to a 5-1 victory in its first Champions League match since the 8-2 loss to Bayern Munich last season.
Navas guessed Bruno's penalty the right way and dived to his left to stop his effort this time, but a video review showed the Costa Rican had both feet off his line when Fernandes struck the ball.
Cavani might have a hard time breaking into United’s team for Tuesday’s game, given his lack of match practice and with Anthony Martial available after a domestic suspension.
Ronaldo is currently self-isolating after testing positive for Covid-19 and will need to pass a negative test before being allowed to feature against Barcelona.
PSG have been without Neymar in four of seven league games this season and the side has managed to win three games.