Philip Billing stunned the Gunners with the second-fastest goal in Premier League history after just nine seconds, and the relegation-threatened Cherries looked like they could pull off a famous victory when Marcos Senesi’s 57th-minute effort doubled their advantage.
Arsenal’s pressure finally paid off five minutes later as Thomas Partey equalised and Ben White levelled matters with 20 minutes still remaining.
Time looked to have run out but Nelson lashed in a superb winner deep into injury time.
Earlier, Phil Foden and Bernardo Silva struck for Manchester City in a 2-0 victory over Newcastle.
Foden opened the scoring at the Etihad Stadium in the 15th minute with a deflected effort after a superb run before Silva came off the bench to secure the points from a feisty encounter.
The Magpies were the dominant side in the early period of the second half and it took the arrival of Silva, who scored within moments of coming on, to settle City’s nerves in the 67th minute.
Newcastle remain four points behind fourth-placed Tottenham, who slumped to a 1-0 defeat at Wolves, Adam Traore netting the only goal in the 82nd minute.
Despite their efforts at the Emirates, Bournemouth ended the day bottom of the table after Southampton earned a priceless 1-0 victory over Leicester.
Carlos Alcaraz scored the only goal in the 35th minute shortly after James Ward-Prowse had missed a penalty, and Leicester were unable to turn pressure into an equaliser, with Harry Souttar heading against the bar in injury time.
Wesley Fofana’s 53rd-minute goal secured a precious 1-0 victory for Chelsea over struggling Leeds, earning under-fire Graham Potter a first win in seven games.
There was no such comfort for West Ham’s David Moyes as they fell to a comprehensive 4-0 loss at Brighton.
Kaoru Mitoma won a penalty that was converted by Alexis Mac Allister in the 18th minute and Joel Veltman chested in a second shortly after half-time before Mitoma and Danny Welbeck piled misery on the Hammers.
Joachim Andersen’s own goal proved enough for Aston Villa to secure a 1-0 victory over 10-man Crystal Palace.
Wilfried Zaha had an early effort disallowed for offside and poor defending culminated in Andersen scoring in his own net in the 27th minute.
Palace’s hopes of a comeback were then dealt a major blow by Cheick Doucoure’s second yellow card in the 62nd minute.
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