Skuse's long-range drive earned the Robins a point when his 29th-minute strike deceived Robert Green, the ball bouncing up and over his diving body.
It means his side have plenty to do if they are to secure automatic promotion to the Premier League.
With Southampton winning at Peterborough, Sam Allardyce's outfit are five points off the top two with just two games remaining.
It could have been so different for the Hammers, who led thanks to James Tomkins' header midway through the opening period and saw second-half chances spurned by Henri Lansbury and John Carew.
But for City, a battling point improves their chances of playing in the Championship next season.
West Ham spent the first 15 minutes camped in the City half, and Ricardo Vaz Te had the ball in the back of the net on the quarter-hour mark, only for referee Craig Pawson to disallow the goal for a handball by Hammers skipper Kevin Nolan in the build-up.
At the other end, Skuse dragged a tame daisy-cutter well wide, while Vaz Te followed suit as neither side could open the other up.
But with 25 minutes gone the away side finally found a way through.
Andre Amougou fouled Maynard wide on the left, Noble floated in the resulting free-kick and Tomkins picked his spot with a free header back across Dean Gerken.
The hosts, already firmly on the back foot, could have been forgiven for folding - but within four minutes they were level.
Skuse latched on to a bobbling ball 25 yards from goal and took aim, more out of hope than expectation.
Green appeared to have the effort covered, but the ball bounced up off the turf in front of him and, although the England stopper got a hand to it, nestled in the right-hand corner of the net.
The Hammers almost nudged ahead just before the break, but Gerken dived bravely on Nolan's shot-cum-cross before Winston Reid could bundle the ball home, and the two teams went into half-time all-square.
The visitors started brightly after the interval, and Lansbury clipped the top of the crossbar before forcing a decent save out of Gerken from a narrow angle.
Allardyce introduced striker Carew for full-back Danny Collins as his side desperately chased a winning goal.
Carew was quickly in the game, shooting low at Gerken after being played in by Vaz Te, but City always looked lively going forward and it required a good stop from Green to keep out Chris Wood's effort with 18 minutes remaining.
Carew headed Matt Taylor's free-kick over and toe-poked wide late on but the Hammers could not find a crucial second.