The 19-year-old scored in the victories against Southampton and Tottenham Hotspur, and saw his performances recognised by the club's supporters via a Twitter vote.
Sterling told the club's official website: "I'm really pleased to have got the award.
"It's been a tough start, but hopefully we can get some wins under our belt and continue going forward like we did at the end of last season.
"The games have been coming thick and fast and we've had to be mentally prepared for each and every one. We're taking it in our stride and hopefully we can get some wins going now."