Cheltenham were then awarded a penalty on 14 minutes when Ben Tozer hacked Russell Penn down. Marlon Pack stepped up and saw his spot kick saved by Lee Nicholls to keep Northampton 1-0 up on aggregate.
The away side then took the lead on 28 minutes through a wonder strike by Luke Guttridge. Akinfenwa laid off the ball once again and it bounced kindly for Guttridge to lash a volley into the back of the net.
Northampton almost doubled their lead on 37 minutes when Chris Hackett's overhit cross nearly went straight into the top corner until the goalkeeper pulled off a fine save.
The Robins pushed forward in the second half hoping to mount a comeback, but Northampton continued to stand firm.
Hackett almost sealed Northampton's place at Wembley on 72 minutes, but his strike was well saved.
Kaid Mohamed almost brought the scores level on 76 minutes, crashing his header off the bar as Cheltenham piled on the pressure.
Northampton held on to secure their place at Wembley.