A foul from Ishfaq Ahmed on Jan Stohanzl earned the hosts a free kick in the 44th minute and Anelka was quick to capitalise. The former Chelsea star's strike curled above the defensive wall to the left post with a helpless goalkeeper, Sandip Nandy, diving in a desperate attempt.
Anelka, who had missed the first three matches due to suspension, showed his class and also earned the much needed three points for his side.
The hosts looked like a new side in the second half, rejuvenated by their one goal lead. They attacked well, seemed eager to add a few more goals to their tally and came close to achieving it a few times.
Mumbai had the chance to double the lead in the 58th minute when Stohanzl skied the shot and frittered away the golden opportunity. Indian forward Subash Singh was also guilty of missing a sitter in the 74th minute.
Anelka did everything right to beat the three defenders and feed the ball to Singh who shot it over the cross bar. Anelka seemed desperate to make up for the lost matches and the French striker came close to adding to his tally by going solo in the 86th minute and the goalkeeper blocked it.
Moritz found the ball on the rebound but couldn't slot it home. The 35-year old had another opportunity a minute later but Moritz squandered the chance inside of the box and went wide.
The hosts looked to increase their lead even in the dying moments of the game when Nadong Bhutia hit the top bar, a few seconds before the close of the match.