The transfer of the 20-year-old had been expected since last week when he spoke openly about his trial with the Gunners, and both clubs are now in a position to confirm that the move has been finalised.
Bramall has moved to North London for an undisclosed fee, widely reported to be in the region of £40,000, with Hednesford expected to receive add-ons should the left-back's move prove to be a success.
The switch marks a remarkable rise for Bramall, who began the season only on trial at Northern Premier League side Hednesford, who play in the seventh tier of English football.
During the first two months of the season, Bramall impressed with his lightning pace down the left flank and he was soon offered a trial with Crystal Palace, where he impressed in a behind-closed-doors fixture.
Sheffield Wednesday were the next team to show an interest and it was during a reserve game with the Owls where Bramall was spotted by Arsenal scout Brian McDermott, who invited him to a two-day trial with the Premier League side.
Bramall had expected to train with the Under-23 squad but he was invited to take part in sessions with the first team, where he did enough to impress Arsene Wenger and his coaching staff.
He will begin life at Arsenal with the second-string but after signing what is believed to be a two-and-a-half year contract at the Emirates Stadium, he is regarded as a potential first-team player for the future.