Jordan Ayew converted penalties in each half and the Black Stars appeared to be cruising to victory with a two-goal lead as the clock passed 90 minutes.
Cape Verde, who had already qualified as group winners, and Egypt drew 2-2 in the other Group B match, which would have put Ghana through with a win.
But goals from Geny Catamo, with the third penalty of the game, and Reinildo Mandava gave Mozambique a point and left four-time champions Ghana on the cusp of a second straight AFCON group-stage exit.
With just two points, four-time champions Ghana will need an unlikely series of results in the four remaining groups to qualify as one of the four best third-placed teams.
Ghana’s exit will be confirmed on Tuesday if Gambia and Cameroon fail to draw in Group C.
The anxiety on the face of Ghana coach Chris Hughton before kick-off was understandable.
A 2026 World Cup qualifying loss to minnows the Comoros last November had heightened calls for the dismissal of the ex-Newcastle United and Brighton and Hove Albion boss.
Losing their opening group match in the Ivory Coast to lower-ranked Cape Verde only added to the pressure on the 65-year-old in his first Cup of Nations.
A draw with record seven-time champions Egypt after leading twice raised hopes that the Black Stars could still progress.
Hughton and his team knew that, at worst, victory over Mozambique would secure third place and probable qualification for the knockout stage.
Those hopes grew stronger after 15 minutes when Palace’s Ayew calmly converted a penalty awarded for a foul on Joseph Paintsil by Nanani.
Ghana suffered a blow in 37 minutes when injured midfielder Majeed Ashimeru was carried off on a stretcher with Iddrisu Baba taking his place.
Hughton also introduced veteran Andre Ayew, older brother of Jordan, before half-time with Paintsil benched.
Realising only a first win in 15 matches spanning five editions would avoid elimination, Mozambique attacked consistently in the second half, but without reward initially.
Midway through the second half the Mambas conceded a second penalty, this time when Atletico Madrid full-back Reinildo handled.
After lengthy protests, Jordan Ayew once again calmly stroked the ball into the net past goalkeeper Ivane Urrubal. Then came the dramatic Mozambican comeback through Catamo and Reinaldo.
