Stevenage continued to dominate and went close to doubling their lead just after half time when Ronnie Henry drove a 35-yard shot against the post.
The home side struggled to make an impact on the game as Boro restricted them to just a few chances.
Forest Green Manager Dave Hockaday told BBC Radio Gloucestershire:
"I'm really disappointed because there was a team out there that wanted it more than us.
"They just kept it away from the danger area.
"We need to be more relentless and tougher, both mentally and physically."
Forest Green: Burton, Hodgkiss, Preece, O'Cearuill (Thorne 46), McDonald (Stonehouse 46), Smith, Rocastle, Brown, Powell, Rigg, Platt (Joyce 79). Subs Not Used: Pass, King.
Stevenage: Day, Henry, Bostwick, Roberts, Laird, Drury (Odhiambo 56), Murphy, Byrom, Boylan (Cole 81), Beardsley, Odubade (Vincenti 73). Subs Not Used: Bayes, Albrighton.
Goals: Boylan 7.
Ref: M Heywood (England).