Dickson Nwakaeme turned the ball into his own net in the 75th, some five minutes after he had replaced Famara Diedhiou who had opened the scoring for the visitors in the second half.
Monaco, who were looking to bounce back from a 4-0 thrashing by neighbours Nice, had equalised through Kamil Glick in the 66th.
The result put the club from the Mediterranean principality on 16 points after seven games, two points ahead of Nice who travel to Nancy on Sunday.
Angers, who are 12th on nine points, took the lead 11 minutes into the second half when Diedhiou latched onto a cross by Toko Ekambi.
Monaco hit back quickly, Glick volleying home from a bad clearance by the Angers defence.
Mathieu Michel then parried a Glick attempt into the path of Nwakaeme, who could do nothing but beat his own keeper.
Olympique Lyonnais's stuttering start to the season continued as they slumped to a 1-0 defeat at Lorient that left them eighth on 10 points.
Paris St Germain suffered their second defeat of the season on Friday, losing 2-0 at Toulouse, and trail Monaco by three points.
(Reporting by Julien Pretot; Editing by Alan Baldwin)
