West Ham cruised to an easy first-half lead after Andriy Yarmolenko took out the entire Rapid defence with an excellent pass to Michail Antonio, who squared it to goalscorer Rice.
Rice looked delighted with a rare goal from defensive midfield, but his celebrations were ruined by some ugly scenes at the near corner flag with a group of police officers and stewards trying to stop any chaos unfolding.
A small number of Rapid fans appeared to have broken over the barrier at the London Stadium, only to be contained by officials, while objects were thrown from both sides.
The scene appeared to calm down relatively quickly, but Rapid supporters continued to intervene in the match, throwing objects at Aaron Cresswell as he lined up to take a corner.
West Ham lead the match 1-0.
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