For much of this game it seemed West Ham's misery would endure. West Ham lost Valon Behrami to injury in the warm-up, saw their captain and inspiration, Scott Parker, suspended for their next two fixtures, squandered a penalty and then conceded what appeared to be the decisive goal five minutes from time when Yakubu headed his first league goal of the season at Goodison Park.
Instead, West Ham did what very few teams in their situation seem capable of – they roused themselves and counter-attacked with verve, with Araujo Ilan meeting Julien Faubert's deep, fast cross with a wonderful diving header to equalise two minutes later.
"I feel fantastic," Zola laughed, pointing to Parker's man-of-the-match champagne. "I am going to get drunk tonight." Since the Sardinian is famously abstemious when it comes to alcohol, this was probably unlikely but he could not stop smiling. It was just a point but, apart from the already-doomed Portsmouth, West Ham were the only team in the bottom seven to have taken one.
"Giving the players three days off was massively important," said Zola. "They had suffered three defeats in seven days and they needed to go somewhere to clear their minds. When they came back we said that, whatever happened, we would fight to the very end. The next five games are not life or death but they are very close."
If this was the start of a belated recovery, it began at a strange venue. Not since December had Everton dropped points at Goodison and, although they were shorn of their usual fluency, it was not a surprise when Diniyar Bilyaletdinov stole in front of Matthew Upson to head past Robert Green.
It was even less of one when West Ham squandered a penalty in extraordinary circumstances. Parker had provided a sublime ball for Carlton Cole, and when the striker was bundled over by Sylvain Distin, David Moyes expected both a penalty and the dismissal of his centre-half. He got one but not the other, although the Everton manager was left aghast at Howard Webb's failure to award a penalty when Louis Saha was all but assaulted by Manuel da Costa in the second half.
There seemed to be a debate between Parker and Mido as to who should take it, and it was the Egyptian who stepped up to deliver the kind of tame spot-kick that most spectators would hope to have bettered in the half-time charity shoot-out.
Along with dressing-room unrest and a disastrous transfer policy, boardroom interference is usually one of the prerequisites for a club to be relegated and West Ham's co-owner, David Sullivan, had arranged Mido's £1,000-a-week contract himself, calling it "an amazing deal". You can say that again. When Webb went to book Mido, the West Ham fans began chanting for their own player to be sent off.
The first equaliser, bundled over the line by Da Costa, was, however, preceded by a delightful chip on to the crossbar from Mark Noble. If the crowd that travelled up from east London had not lost their sense of humour, their players have not entirely lost their skill.
Everton: (4-1-4-1) Howard; Neville, Jagielka, Distin, Baines; Heitinga; Bilyaletdinov (Yakubu, 65), Osman (Rodwell, 83), Cahill, Pienaar; Saha. Substitutes not used: Turner (g), Hibbert, Senderos, Duffy, Wallace.
West Ham United: (4-4-2) Green; Faubert, Da Costa, Upson, Spector; Stanislas (Gabbidon, 89), Kovac, Parker, Noble; Mido (Ilan, 77), Cole. Substitutes not used: Kurucz (gk), McCarthy, Daprela, Spence.
Referee: H Webb (South Yorkshire).
Booked: Everton Distin, Cahill; West Ham Parker, Noble, Mido.
Man of the match: Parker