Jamilu Collins' cross from the left ought to have been simple for Moez Ben Cherifia to collect, but the goalkeeper collided with his own defender Yassine Meriah, allowing Ighalo to pounce for his fifth goal of the competition.
Algeria play Senegal in the Cup of Nations final in Cairo on Friday.
It took 57 minutes for them to have their first effort on target, a whole 15 minutes longer than their entire match against the Super Eagles lasted - after the last third-place play off between these sides ended with Tunisia walking off in protest in the 42nd minute.
Grisha was in charge of two matches of the Super Eagles during the 2018 FIFA World Cup qualifying campaign: the 2-1 defeat of Zambia in Ndola in October 2016 and the 4-0 defeat of Cameroon in Uyo in September 2017.
Alain Giresse has a well-drilled squad, but we just fear a lack of star quality may cost them in a more open contest for the bronze medal.
The 36 year-old, who debuted on the worldwide scene eight years ago, recently served as fourth official in Nigeria's AFCON 2019 semi-final defeat to Algeria.
"The statistics were even better for us, because we had 11 shots on goal, they had only six".
"Though third-place is a consolation for us, but it is becoming too much".
For three-time champions, Nigeria outwitted newcomers Burundi 1-0. It was in 1962 against Uganda (3-0).
Coach Rohr said the loss was "tough to digest" for his players but stressed the current crop of Super Eagles had the potential to become "very good" given time.
Both teams had earlier missed penalties, while the Eagles of Carthage had another spot-kick ruled out by VAR after Salif Sane headed the ball against Idrissa Gueye's arm. The striker has been involved in 63 per cent of Nigeria's goals in this edition (five of eight).