The problems – and consequently the pressure – are mounting for Luiz Felipe Scolari. A longstanding struggle to break down opponents at Stamford Bridge has been exacerbated by a failure to hold out in matches they have dominated, with Clint Dempsey’s late header giving Fulham a point that they deserved for their persistence, if not for the quality of their football.

As a gentleman, Scolari had the good grace to wish the media a happy new year after he had rounded on his players’ poor defending in the postmatch press conference, but the Chelsea manager’s own outlook is far from rosy. A run of disappointing results has hardened from a blip to a trend, with his team collecting only ten points from the past 21 available, winning only two of seven matches.

Fulham are in the unusual position of being the form team in West London, with a nine-match unbeaten sequence enabling Roy Hodgson to celebrate his first anniversary as manager of the club in style. Finishing 2008 ninth in the Barclays Premier League – ten places higher than when he took over in January – should make Hodgson a genuine candidate for any award for manager of the calendar year.

Scolari’s stock, by contrast, is falling. Chelsea fans questioned his judgment with shouts of “You don’t know what you’re doing” midway through the second half. That the same people were celebrating Frank Lampard’s second goal, from a free kick, moments later demonstrates the fickle nature of the modern supporter, without completely detracting from the legitimate doubts surrounding the Chelsea team. The invincible swagger of the early José Mourinho era is a thing of the past.

Scolari was right to criticise the defending that allowed Dempsey to leap above Didier Drogba and head the equaliser from Simon Davies’s corner after Dempsey had given Fulham the lead in the tenth minute, though as such shambolic marking has been a feature of their season, the manager must also look at himself.

Fulham’s first goal was even softer, with Davies’s free kick sailing past four blue shirts on its way to the far post, where Dempsey was allowed to flick the ball past Petr Cech. No one took responsibility for clearing the ball in the manner that John Terry would have, raising further questions about Chelsea’s strength in depth in the absence of their captain and Ricardo Carvalho.

Bordeaux, Arsenal and West Ham United have scored from set-pieces against them in the past month, with Chelsea suddenly surrounded by an air of vulnerability that would have been unthinkable under Mourinho. Some hard work on the training ground is in order in the next few days. Scolari is deserving of some sympathy, however, as several of his key players continue to let him down.

Florent Malouda and Alex limped off in the first half with hamstring injuries after showing precious little appetite for the game, though the latter’s removal did at least facilitate the return of Carvalho from the bench after three months out with a knee problem. The Portugal centre back will be crucial if Chelsea are to challenge for honours in the coming months.

Whereas Carvalho slotted back in seamlessly, Deco was anonymous on his return to the side, with Barcelona’s surprising decision to let him leave last summer looking more sound by the day. Scolari’s persistence with him, by contrast, is looking increasingly puzzling, with the Chelsea fans showing their derision when Joe Cole was hauled off instead after another hard-working display.

Drogba also showed his battling qualities in bringing several smart saves from Mark Schwarzer and Nicolas Anelka did his best despite being isolated out on the left, but without the sterling contribution of Lampard, Chelsea could easily have lost.

The England midfield player provided a true captain’s performance, driving his colleagues on and scoring two typically opportunist goals, his tenth and eleventh of what is shaping up to be another 20 goals-plus season. Unfortunately for Fulham, the otherwise outstanding Schwarzer was culpable for both, colliding with Aaron Hughes to allow Lampard to tap in from close range in the 50th minute and then being wrong-footed by his free kick in the 72nd.

Chelsea looked like going on to win comfortably at that stage, which would have been harsh on the likes of Dempsey and Andrew Johnson, but such certainties can no longer be taken for granted. This was the fifth occasion when they have led and failed to win this season, a weakness that is unlikely to be rewarded with trophies in May.

Fulham (4-4-2): M Schwarzer 6 J Paintsil 6 B Hangeland 6 A Hughes 6 P Konchesky 5 S Davies 6 D Etuhu 5 D Murphy 6 C Dempsey 6 A Johnson 7 R Zamora 6 Substitutes: L Andreasen (for Etuhu, 79min), E Nevland (for Zamora, 79), C Baird (for Murphy, 90). Not used: P Zuberbühler, J Gray, F Stoor, T Kallio. Next: Blackburn (h).

Chelsea (4-1-3-2): P Cech 6 J Bosingwa 6 Alex 5 B Ivanovic 5 A Cole 6 J O Mikel 6 J Cole 7 F Lampard 8 Deco 5 F Malouda 5 D Drogba 6 Substitutes: N Anelka 6 (for Malouda, 31), R Carvalho 6 (for Alex, 37), S Kalou (for J Cole, 72). Not used: C Cudicini, M Ballack, P Ferreira, J Belletti. Next: Man United (a).

Referee A Marriner Attendance 25,462