Sri Lanka made 248-9 after half-centuries from Kusal Mendis (53), Dinesh Chandimal (62) and captain Angelo Mathews (56), as Liam Plunkett took 3-46.
Rain delayed the start of the run chase and England were 16-1 as the weather forced the abandonment at 17:30 BST.
England lead the five-match series 1-0.
Alex Hales, who scored 133 not out in the second ODI, was out for a golden duck to the third ball of the run chase, leaving fellow opener Jason Roy and Joe Root at the crease when play ended. (Courtesy BBC)