Rovers dominated the opening 20 minutes and they created a clear opportunity in the seventh minute when Steve Perrin's goal kick was flicked on by Kayode Odejayi and Neil Grayson's attempted lob was tipped wide by visiting keeper Aaron Kerr.
Three minutes later Forest Green took the lead when Grayson's free kick was blocked and Kevin Langan's cross back into the box was flicked on by Jon Richardson for Grayson to fire it into the top right-hand corner of the net.
They almost increased their lead just before the half hour when Martin Foster's cross from the right wing found Alex Meechan racing into the back post but his header went inches wide.
The visitors almost equalised eight minutes before the interval when Bobby Howe's pass split the Rovers defence wide open and James Taylor's diving header needed Perrin to tip the ball away at full stretch.
Forest Green went from playing well in the first half to the complete opposite in the second and it was no surprise when the visitors equalised after a simple ball in to Forest Greens penalty area was not cleared by the defence and Paul Wood tapped the ball over the line form close range.
Havant fired their way into the semi-finals late on when Steve Jenkins' mistake allowed Tim Hambley to volley past the helpless Perrin into the top of the net.