It was going so well for Everton until the final 11 minutes and then it all went wrong as they conceded two penalties and had their defensive strong man Slaven Bilic sent off.
Howard Kendall will truly believe that fortune was cruel to his side as they fought courageously to climb away from the foot of the table. There was no arguing with the first penalty as Mitch Ward upended Roberto Di Matteo, enabling Dennis Wise to take advantage, but Bilic's double punishment after he challenged Tore Andre Flo appeared harsh. This time Gianfranco Zola drove home the penalty.
Kendall did not need a visit to Stamford Bridge to remind him of the difficulties he has faced in supplementing his meagre attacking resources with a big new signing. Chelsea were in the comfortably contrasting position of being able to leave out both Mark Hughes and Flo as they gave Gianluca Vialli a first start in the Premiership for two months.
In defence Michael Duberry reappeared after an ankle injury had sentenced him to a similar sideline wait. He never strayed far from the side of Duncan Ferguson and a couple of successful early challenges gave Chelsea a compact look.
Bilic and Craig Short were similarly important in Everton's defence and a vital tackle from the latter, dispossessing Vialli on the edge of the 18-yard line, helped his team to settle.
They would have settled better had Gareth Farrelly not shamelessly wasted a splendid opening created by Ferguson's reverse pass. The attacking midfielder, promoted in place of the departed Graham Stuart, ran strongly behind the Chelsea defence but failed even to bring Ed de Goey to a save, screwing his shot wide of the near upright.
Filling the right wing-back role for Everton was Ward, part of the deal that sent Stuart to Sheffield United, with Carl Tiler also arriving at Everton.
Zola, recently out of form and out of goals, bent a free-kick over the wall but Ferguson came to Neville Southall's aid by the post. It signalled an improved phase from Chelsea.
Chelsea needed more width to truly stretch their opponents. Just before the interval Vialli opted to take the ball wide when found by Wise but, when he pulled it back across the face of the goal, there was nobody there.
The second half did not begin well for Ferguson, who saw a well-won header from Ward's cross smash against De Goey's left-hand post and bounce back into play. Kendall, cursing his luck and shooting an angry look towards the skies, was a study in abject frustration.
The near miss was still troubling Ferguson six minutes later when he hurled Wise to the floor after the Chelsea captain was late in the tackle. The Scot was booked as Wise had just been.
Chelsea sent on Hughes and later Flo to give themselves more strength and height in the last third but their main threat came now from Frank Sinclair's more frequent right-wing sorties. Everton, sensing that a point was theirs for the taking, pulled back defensive reinforcements to keep them at bay. They succeeded in that aim until the 79th minute.
Chelsea: De Goey, Petrescu, Babayaro, Clarke (Granville 51), Vialli (Flo 75), Wise, Duberry, Di Matteo, Sinclair, Nicholls (Hughes 51), Zola. Subs Not Used: Hitchcock, Crittenden. Booked: Wise. Goals: Wise 80 (pen), Zola 90 (pen).
Everton: Southall, Ward, Hinchcliffe (Oster 83), Williamson, Short, Bilic, Phelan (Ball 68), Barmby, Ferguson, Speed, Farrelly (Cadamarteri 83). Subs Not Used: O'Toole, Thomas. Sent Off: Bilic (90). Booked: Hinchcliffe, Short, Ferguson.
Referee: N Barry (Scunthorpe).