The awards follow something of a renaissance for Manchester United, who bounced back from losing their opening two Premier League games to win their next four.
The month of September suffered a slight fixture disruption following the death of Queen Elizabeth II as a full round of Premier League action was called off as a sign of respect to the long-serving monarch.
Most teams played two round of matches during the month.
Ten Hag beat off competition from Gary O'Neil and Antonio Conte to win the manager of the month.
Rashford scored two goals and provided two assists in September and saw off Philip Billing, Pierre-Emile Hojbjerg, Kevin De Bruyne, Jacob Ramsey and Alex Iwobi to win the award for the first time since January 2019.
"My favourite moment of September was an assist actually," Rashford said after receiving his award.
"Antony's first goal, on his debut. It was a massive moment and it gives us strength and courage. Seeing him come into the team and score right away was massively important."
"It's a good feeling. We didn't start the season so strongly so to get four wins on the bounce is really good.
"It's still fairly new with the manager and we've got much better at doing the things he wants us to. But there are still a lot of steps to take and we're all looking forward to doing that together."
Manchester City striker Erling Haaland won player of the month for August.
