Mathew Stevens gave the hosts the lead on the half-hour mark before Omar Bugiel scored Sutton's first-ever league goal in the 69th minute.
The match looked to be headed for a draw until Adams headed home at the death to give the Green Army all three points.
Story of the match
Sutton had the first chance of the match in the third minute as David Ajiboye met the corner of Robert Milsom, but his header was wide. Donovan Wilson then had a go with a snap shot, but he also shot wide.
Jamile Matt had the hosts' first really good chance as he got his head to a cross in the six-yard box, but the Sutton defense cleared the danger. Forest Green then saw a free-kick by Harry Beautyman deflected off the wall and over the net.
On the half hour mark, the Green Army took the lead as Matty Stevens rose highest to loop a header past Yellows goalkeeper Dean Bouzanis to make it 1-0.
Four minutes into the second half, Wilson thought he had leveled only to be called for a foul. Beautyman then played in a lovely ball to Wilson, only to see his flick saved by Luke McGee.
The action was now end-to-end, Adams firing over while at the other end, Wilson fired across the box from a wide angle with no luck while Bouzanis got a fist to a Nicky Cadden corner.
Ajiboye was provider for the Yellow Army's equalizer in the 69th minute as he crossed to Bugiel, who headed past McGee in the top right corner for Sutton's first-ever league goal.
Forest Green went in search of a winner and a cross just evaded Bouzanis and comes off of the top of the bar. Ajiboye then tried his luck from the edge of the box, but went over.
The points looked to be shared, but Adams provided a late twist as in the first minute of stoppage time he headed home from a free-kick to consign a brave Sutton team to defeat on their Football League debut.
Man of the match: Ebou Adams
Seeing his team outplayed for the majority of the match, the Gambian provided a late moment of magic to secure Forest Green all three points.