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However, Joao Gomes received a second yellow card for a studs-up lunge on Jurrien Timber in the second half, denting hopes of a result for the hosts.
They could not hold out as Calafiori was on hand in the 74th minute to give Arsenal a massive victory, keeping them six points behind Liverpool, who beat 18th-placed Ipswich Town on Saturday.
Fourth-bottom Wolves, meanwhile, are only above Ipswich on goal difference, and could drop into the relegation zone when Leicester City visit Tottenham Hotspur on Sunday.
How the match unfolded
Despite Arsenal bossing possession, Wolves created the first big chance as Nelson Semedo crossed for Pablo Sarabia, who miscued his volley over.
Kai Havertz then missed two big chances, nodding just past the post before seeing another close-range header brilliantly kept out by Jose Sa’s feet.
Wolves got a huge boost on the stroke of half-time as Doherty looked to counter from an Arsenal corner, only to be felled by Lewis-Skelly. Referee Michael Oliver produced a red card for serious foul play.
The hosts then should have gone ahead in the 64th minute. Moments after being denied by a sprawling David Raya, Matheus Cunha prodded agonisingly wide.
But the momentum turned when Gomes jumped into a senseless challenge on Timber high up the pitch, receiving a second booking and giving Arsenal the impetus to press for a winner.
They got the decisive goal in the 74th minute, Calafiori arriving at the far post to drill a low effort beyond the despairing dive of Sa.
Wolves miss golden chance
Going into half-time goalless and with a one-man advantage, Wolves found themselves with a great opportunity to cause an upset and pull three points clear of the danger zone.
Pereira’s men had managed to keep Arsenal’s full complement at bay in the first half, with Emmanuel Agbadou particularly impressive on his home Premier League debut, using his strength and speed to keep Gabriel Martinelli quiet on the Gunners’ right flank.
However, Wolves showed little in attack after Sarabia’s early miss, with Cunha again cutting a frustrated figure just days after his body language was criticised by Pereira following a 3-1 defeat at Chelsea.
If an expectant Molineux crowd was expecting more in the second half, they were sorely disappointed as Arsenal made much of the running even with 10 men, Havertz missing another good headed chance before Wolves finally created a pair of openings for Cunha.
But just as the momentum seemed to be with the hosts, Gomes evened things up with his needless lunge on the edge of Arsenal’s box.
To make matters worse, striker Jorgen Strand Larsen limped off clutching his hamstring in the first half. Daunting clashes with Aston Villa and Liverpool are next up for the strugglers.
Defenders lead the way for Arsenal
Coming into Saturday’s match, Arsenal had scored seven Premier League goals via defenders this season – more than any other team. Wolves, meanwhile, had conceded 13 goals to defenders, also a league-high figure.
Mikel Arteta must have been fearing the worst when Lewis-Skelly was sent off, with Arsenal having failed to win any of the previous three matches in which they had received a red card this season. The Gunners battled to draws with Brighton & Hove Albion and Manchester City, but they were beaten at AFC Bournemouth when William Saliba was sent off in October.
Arteta also saw his forwards misfire once again, with Havertz missing three excellent headed chances, the best of those drawing a reflex save from Sa midway through the first half.
But Arsenal’s defenders led the way once again, with Calafiori netting his second Premier League goal at a crucial time.
Calafiori’s first strike in red helped Arsenal earn a draw at the Etihad Stadium in September, but this one could prove even more important as they kept pace with Liverpool. Another meeting with Pep Guardiola’s City is next up in eight days.
Match officials
Referee: Michael Oliver. Assistants: Stuart Burt, James Mainwaring. Fourth official: Tony Harrington. VAR: Darren England. Assistant VAR: Adrian Holmes.