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Nuno took charge of his first match as manager of the Hammers just 48 hours after being appointed to replace the sacked Graham Potter.
Former Nottingham Forest manager Michael Keane gave Everton a first-half lead at Dickenson Hill, but international striker Jarrod Bowen spared Nuno's defeat in his first match with the London club with an equaliser in the 65th minute.
This was only the second time in seven matches in all competitions this season that West Ham had managed to remain unbeaten.
The East London club is 19th in the league, second from bottom, with four points from their first six matches.