The Green Lions took the lead through Matt Loaring on 26 minutes, a header that moved him clear as the club’s second highest scorer with 93 goals.
Thatcham went down to 10 men when Callum WIllmouth picked up his second yellow, leaving GFC in a strong position at the break.
Both sides then had their chances.
Ross Allen put a 30 yard free-kick onto the crossbar, before Solway cleared off the Green Lions line.
In the 79th minute the game changed again with the sending off of Dave Merris for a second yellow.
Just a minute later, Thatcham pounced on a loose pass and Owen Clark scored.
But this GFC side are used to leaving it late to settle matches.
With just two minutes of normal time to go, Ross Allen rose to head home and then from the restart Charlton Gauvain drove down the middle of the pitch and scored to settle the match.
GFC climbed one place to 18th with the win as their quest to pull clear of the relegation fight continues.
They are at home to Raynes Park on Wednesday with the kick-off at 19:45.