Off the pitch the Hammers have been troubled with an administrative error which led to the resignation of the club secretary and means they have to replay their Worthington Cup quarter-final with Aston Villa.
It has been difficult on the pitch too, and this result leaves them with only two wins in 11 Premiership matches. No wonder they started this game like a team short on confidence.
However, Derby are in the relegation zone and came with only one win in their previous eight games, yet they belied that form after just four minutes. Darryl Powell broke into the penalty box and a magnificent sliding tackle by Rio Ferdinand on the midfielder only saw the ball fall to an unmarked Dean Sturridge, who sidefooted past Shaka Hislop.
Powell was a notable force in the early exchanges, getting the better of his opposite number Joe Cole, but once West Ham took a grip on midfield through Marc-Vivien Foe and Frank Lampard, the teenage prodigy Cole began to show his range of passing.
Within five minutes of taking the lead, Derby's Horacio Carbonari cleared off the line from Foe's close-range shot, but it was not long before the home side drew level in spectacular style through Paolo Di Canio, who signed off the footballing century with an effervescent performance.
With 21 minutes gone, Scott Minto passed to the Italian and from an unpromising position inside the penalty box he turned his marker and curled an unstoppable shot into Mart Poom's top left-hand corner.
Derby had done well to hold out for that long, given that a flu virus had deprived them of four players, while a groin injury had ruled out their recent record signing, the striker Branko Strupar. To further dent their confidence, they had lost 2-0 to Aston Villa on Boxing Day.
West Ham, who drew with Wimbledon then, were at near full-strength, including Paulo Wanchope against his old side, yet the Rams' defence was able to keep West Ham at bay for long stretches, forcing Lampard into long-range efforts just before half-time that Poom was able to watch go wide. However, Poom was tested to the full just two minutes before the break when a fluid move saw Wanchope flick the ball on to Di Canio, whose snap volley from 12 yards was smartly pushed over by the Estonian.
Yet the Hammers' defence when tested appeared shaky, and with 25 minutes gone Sturridge nearly restored his side's advantage after the ball rebounded for him inside the area, but Hislop charged his shot down.
The second-half momentum was with the home side as Derby defended grimly in the face of an East End onslaught that threatened danger but rarely achieved any. Two minutes after the break, Cole shot from 20 yards but his effort flew over the angle of post and crossbar. Constant possession by the Hammers failed to yield the winner, although they again went close with three minutes remaining. A Trevor Sinclair cross was met by Lampard but his strong shot was brilliantly diverted to safety by Poom, a save his manager Jim Smith described as "unbelievable".
Goals: Sturridge (4) 0-1; Di Canio (21) 1-1.
West Ham United (3-5-2): Hislop; Potts, Ferdinand, Margas; Foe, Lampard, Cole, Sinclair, Minto (Keller, 60); Wanchope, Di Canio. Substitutes not used: Forrest (gk), Carrick, Coyne, Newton.
Derby County (3-5-2): Poom; Elliott, Carbonari, Powell; Laursen, Johnson, Bohinen, Burley, Prior; Robinson (Beck, 82) Sturridge. Substitutes not used: Hoult (gk), Burton, Nimni, Jackson.
Referee: A Wiley (Staffordshire).
Bookings: Derby County: Sturridge, Prior, Burley, Laursen. West Ham: Wanchope, Ferdinand, Sinclair.
Man of the match: Poom.