Ayew broke the deadlock with a glancing header from Mathieu Valbuena's corner in the third minute of time added on leaving the Inter players shaking their heads in disbelief as the French side stretched their unbeaten run in all competitions to 16 games.
Inter coach Claudio Ranieri left Argentine striker Diego Milito on the bench, even though he is the only player to have scored for the team in their last seven games, netting all the goals in the 4-4 home draw against Palermo in Serie A three weeks ago.
Wesley Sneijder and Mauro Zarate were unimpressive against Marseille but Milito was never called upon as Ranieri used only two of his three possible substitutions with Giampaolo Pazzini also staying on the bench.
"Milito had influenza for a week. I did not want to risk him," Ranieri told reporters.
"Marseille were pushing on the left flank and Joel Obi did a good job to give us more balance."
Ranieri appeared unconcerned by the task facing the Nerazzurri in the return leg at the San Siro on March 13.
"The San Siro will be full and it will be an advantage for us," he said. "We can score two goals no problem.
"We played well, we had good counter-attacks, we had more opportunities and they scored on their first chance.
"We did not deserve to lose," Ranieri added.
'SMALL ADVANTAGE'
Marseille coach Didier Deschamps was cautious about his team's chances of making the quarter-finals.
"It's a small advantage. It's going to be complicated," he told reporters. "We will not be in a position to only sit back and wait."
Inter's best chance of the match came in the 11th minute when Steve Mandanda tipped a Diego Forlan half-volley over the bar.
The hosts enjoyed plenty of possession but were made to wait until the half-hour to threaten Julio Cesar when a free-kick landed in front of goal only for Brandao and Souleymane Diawara to get muddled up and put the ball out of play while the Inter defence went AWOL.
The unmarked Zarate had a chance for Inter eight minutes before the break but his low shot was too weak to bother Mandanda.
Inter looked dangerous on the break but the Marseille defence held a high line and increased the pressure in the second half, although they struggled to create clear chances.
Obi replaced Zarate shortly after the hour as Ranieri looked to add fresh blood to his front-line.
His Marseille counterpart Didier Deschamps replaced Brandao with Jordan Ayew in the 73rd minute but the Ghana forward produced an equally unimpressive performance.
Just when it looked as though Inter were going to keep a rare clean sheet, Ayew, who recently returned from the Africa Cup of Nations where he helped Ghana reach the semi-finals, headed home with the last move of the game.
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