The youngster will join the club's U-23s squad and will train under coach Steve Gatting.
The club confirmed the news on their official website as well as on their official twitter handle.
Their official statement read: "We have signed young left-back Cohen Bramall from Hednesford Town in the Northern League Premier Division.
"The 20-year-old, who impressed on a recent trial, will join up with Steve Gatting and our under-23s squad.
"Cohen has come through non-league football and played with Market Drayton last season.
We'd like to welcome Cohen to the club."
Their official Tweet read:
Wenger lauded Arsenal's newest recruit as a promising signing. Speaking to arsenal.com the manager said: "I believe that when you go down, there's mental selection.
"Are you ready enough to fight [to show] that you are above this level and that you absolutely want to come back? Players like Giroud and Koscielny have had that test. They are similar to that. Once they get up, they really mean it.
"Bramall is a young, promising left back who has a lack of experience at the top level but who has fantastic ingredients.
"He has tremendous pace, a good left foot, a great desire to do well. Overall, he's a very exciting prospect."
Arsenal are currently at the fifth position in the league table. They have 41 points from 20 matches.