After 120 minutes of scoreless action, goalkeeper Abou Gabal saved spot-kicks from Harold Moukoudi and James Lea Siliki before Clinton N'Jie completely missed the target.
Egypt will take on Senegal in the showpiece on Sunday, though Carlos Queiroz will not be in the dugout after being sent off for two displays of dissent towards the end of normal time.
In a low-tempo game littered with mistakes, the best chance of the first half came when Michael Ngadeu-Ngadjui saw his header hit the post in the 18th minute.
Mohamed Salah was presented with a glorious chance to open the scoring in the second half by a short back pass from Martin Hongla, but he was unable to round Andre Onana after the goalkeeper hared out of his box to intervene.
Samuel Oum Gouet went close to scoring a goal of the tournament contender when his rasping 35-yard drive clipped the outside of the post, and Queiroz was sent off before the start of extra time.
The additional 30 minutes were not enough to separate the sides, but Abou Gabal's heroics sent Egypt through to the final after they missed out on home soil in 2019.