Ayew scored with the last touch of the game to inflict a fourth straight defeat on the Italian side and pile the pressure on coach Claudio Ranieri.
Inter were improved from the side that went down 3-0 to Bologna on Friday for their third straight home defeat, but still looked sluggish at times and there were clear signs of anxiety as their woeful run continued.
Ayew headed in from a corner he had won in the third minute of injury time to make it six defeats out of seven for the Italian giants.
Although a 1-0 scoreline by no means eliminates Inter as they head back to the San Siro for the second leg, Ranieri must wait to see if his paymasters allow him to stay in charge for that match.
There were early chances for Marseille as Inter began nervously.
Benoit Cheyrou saw an early effort deflected narrowly wide for a corner before sending a free-kick from a good position straight into the wall after Mauro Zarate conceded a soft foul on the edge of the area.
The Italians began to find some attacking purpose, though, and Steve Mandanda made an excellent save to push Diego Forlan's close-range volley over the bar and the goalkeeper was on his toes again to beat Zarate to the ball at the near post from Javier Zanetti's low cross, timing his intervention to perfection.
Wesley Sneijder's free-kick struck the wall but Inter quickly gathered the loose ball and Zanetti's pass struck Esteban Cambiasso, spinning inches wide, although the assistant had wrongly signalled offside.
Marseille could and perhaps should have scored moments later when Mathieu Valbuena sent a curling free-kick into the area only for Brandao and Souleymane Diawara to get in each other's way and send it wide with the goal gaping.
Inter countered with their best move of the match as Schneider sent a defence-splitting pass through to Cambiasso. He pulled it back for Zarate, but the striker could get no power on his shot from a good position.
Marseille came out of the dressing room for the second half looking like the better side, but Diawara's long volley went wide while Julio Cesar made a comfortable enough save from Cesar Azpilicueta drilled effort from the edge of the area.
A handball from Diawara gave Inter a set-piece to work with, but Dejan Stankovic fired wide from Schneider's lay-off. They tried a similar move moments later, but this time Stankovic's shot found Mandanda.
Jordan Ayew had a shot charged down by Cambiasso before his brother Andre set off on a one-man mission to win it for Marseille late on.
He headed Rod Fanni's cross over the bar but got better contact on another from Morgan Amalfitano, only to send the ball into the arms of Julio Cesar when placing it either side would have made for a certain goal.
It was the elder Ayew again in injury time, cutting in from the left wing and then trying a curling shot off the outside of his boot which Julio Cesar could only palm aside for a corner.
Valbuena swung it in, and Ayew was on hand to nod it into the far corner with Cristian Chivu pawing helplessly at his shirt.
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