The Portuguese, who has also been fined $27,000, was sent off after kicking a bottle during United's Premier League clash with the same opponents on Monday (AEDT), which finished 1-1.
Mourinho's angry outburst was in reaction to Paul Pogba being cautioned for a dive by referee Jon Moss.
It is the third time Mourinho has fallen foul of the FA this season - he was fined $85,000 for comments made about the appointment of Anthony Taylor to referee United’s clash with Liverpool in October, and was also sent off during United's 0-0 draw with Burnley at Old Trafford later that same month.
That dismissal resulted in a one-match ban and an $13,500 fine.
An FA statement on Thursday confirmed Mourinho would serve his latest suspension with immediate effect, the announcement coming three hours before the clash with the Hammers was due to kick off.