Fulham dominated the start and worked the ball patiently until Antonee Robinson released Palacios to the byeline, where he swept in a cross for Muniz to head home the Cottagers' first away goal of the season from six yards.
The lead lasted three minutes as Konstantinos Mavropanos headed the ball across for Morato to power a header into the net.
Fulham struck again in the 37th minute when Joachim Andersen's long pass sent summer signing Palacios clear to lift the ball over Finlay Herrick for the Spaniard's second goal in successive games.
But West Ham showed great fight and levelled again on the stroke of half‑time as keeper Benjamin Lecomte parried Kyle Walker‑Peters' cross into the path of Pablo, who squeezed in from a tight angle.
Fulham boss Arbeloa, who made nine changes from the side that drew 0‑0 at Liverpool on Saturday for a first league point after three defeats, made good use of his substitutes - not least the introduction of match‑winner Bobb in the 69th minute.
The Norwegian, who arrived from Manchester City in January, enjoyed his first goal since netting at the Fifa Club World Cup in June 2025 for his former club.
Both sides had late chances, with Taty Castellanos' 88th‑minute effort cleared off the line by Timothy Castagne and Palacios dragging a shot wide of an open goal.
West Ham's attention will turn to the league and visit Millwall next in the Championship on Saturday (12:30 BST), while Fulham will look for their first Premier League win when they host Manchester United on Sunday (16:30 BST).