The 16-year-old midfielder scored twice against Arsenal before adding another goal against Fulham as Cobham Next Gen ended the month in fourth place.
Watson opened September with a brace in a 4-1 victory over Arsenal. He converted a penalty before adding a free-kick that struck the post on its way into the far corner.
He then made another decisive contribution against Fulham, scoring from distance to put Chelsea ahead after the visitors had taken an early lead.
Chelsea confirmed the award after Watson topped an eight-player shortlist selected by the Premier League's Football Development Panel.
The award recognises a month in which Watson scored three times in two Premier League 2 appearances. His goals came in consecutive London derbies and helped the development side collect two wins.
Watson has also made a strong start to the wider campaign, finding the net in each of Cobham Next Gen's opening four PL2 matches.
His progress has extended beyond academy football. Earlier in September, Watson made his competitive senior debut when he started Chelsea's 6-3 Carabao Cup victory over Leeds United at Stamford Bridge.
The teenager is the son of former Crystal Palace, Wigan Athletic and Watford midfielder Ben Watson and has already attracted attention for his performances during the early weeks of the season.
His September award adds another marker to that progress after a month in which he combined regular goals for the development side with an opportunity at senior level.
Watson's three September goals were enough to secure the monthly award, adding individual recognition to a campaign in which he has already established himself as a regular scorer for the club's development side.
The monthly award also follows his first senior appearance, giving his September an extra dimension. His development has therefore continued on two fronts, with academy performances producing both goals and recognition while senior opportunities begin to arrive.
For Cobham Next Gen, Watson's scoring run has provided a consistent attacking contribution at the start of the new campaign. The award now gives the midfielder an individual accolade to match that early run of form.