The news was announced on Wednesday by the Algerian football federation (FAF), which was fined $100,000 (£74.5k) for security lapses and improper conduct by players, officials and supporters.
Goalkeeper Zidane and defender Belghali will serve their suspensions during qualifying for the 2027 tournament.
The FAF said it was lodging an appeal to challenge the sanctions.
Algeria were denied a penalty with the game locked at 0-0 before they eventually lost 2-0 to Nigeria thanks to second-half goals by Victor Osimhen and Akor Adams.
After the final whistle a scuffle broke out between both sets of players and staff. The FAF said the fines included charges for fans attempting to invade the pitch and displaying bank notes towards match officials.