The match at Newfield went right to the wire, with only two late penalties from Academy captain Fraser Gillies edging the home team in front and putting them through to a clash on 11 February with Stewart's Melville College at Murrayfield.
The under-18 team have not made it to a final for over a decade and Edinburgh Academy coach Mark Appleson said: "It was a brilliant match watched by a big crowd and this is what the cup is all about. The pitch was tricky, but the boys tried to play good rugby when they had the ball in hand and eventually it saw us through.
"Fraser (Gilles] had missed a couple of kicks earlier in the match, but he is as calm as they come and when we were 15-13 down I thought if we got chances then he would kick them over. Thankfully I was proved right and the boys can now start preparing for what will be a big night for them at Murrayfield.
"I thought our scrum-half and stand-off played well, but for me the man of the match was our second-row Henry Binnian. He took every kick-off, was solid in the lineout and gave us a platform to build from."
The home side made a quick start to the match, but it was the visitors – looking to get to the final for the second time in three years – who got the first points on the board.
Some quick thinking after a penalty was awarded saw stand-off Scott Mullins go over. The same player kicked the conversion and a penalty to extend the lead to ten points, but the home team were soon on the scoreboard.
Strong-running centre Ben Brown scored a try and by the break they had evened things up thanks to the boot of Scotland under-18 squad man Gillies.
The stand-off kicked another penalty ten minutes after the break to give his side the lead for the first time before George Watson's scored the try of the game.
Mullins chipped a deft kick into the arms of prop Chris Scott out on the wing and he showed good strength and composure to dive in at the corner.
Academy did not panic despite being behind once again and a lot of credit must go to their pack, led by Binnian, for keeping going.
Their endeavours were rewarded when the visitors were penalised twice late on. Gillies kicked both penalties and although Watson's then went frantically on a search of a third try it was not to be and it is Edinburgh Academy who will be playing at the national stadium in two-and-a-half weeks' time.
Scorers: Edinburgh Academy: Try: Brown. Pens: Gillies 4. Con: Gillies. George Watson's College: Tries: Mullins, Scott. Con: Mullins. Pen: Mullins.
Edinburgh Academy: A Simpson; A Normand, B Brown, M Zegleman, T Whittle; F Gillies, J Munro; S Moodie, S Murphy, R Campbell, D Forbes, H Binnian, N Plevris, L Seydak, C Dean. Subs used: D Falkner, J McLean.
George Watson's College: S McKean; G Dalkin, A Chalmers, J Ferguson, R Young; S Mullins, A Harris; C Scott, L Reid, G Marr, D Marinello, R Davies, M Irvine, I MacLennan, M Peacock.
Referee: A Hoyle.