The all-important goal was scored by defender Subhasish Bose in the 46th minute from a penalty.
Sporting Clube de Goa, who beat Mumbai FC in the away match by identical margin, will now meet Aizawl FC in the first semi-final on May 7.
The match witnessed a disappointing start as neither of the two teams were able to impress in the barren first half.
Sporting had the first real chance after half an hour Sumeet Passi's shot from a Ganesh Thakur pass was cleared by a rival defender.
Rowllin Borges, who controlled the midfield well, twice stretched Mumbai FC keeper Kunal Sawant to make good saves.
The visitors, who failed to impress in the first half, had a early chance after the change of ends but Taisuke Matsugae saw his header going over the bar after Minchol Son relayed the ball to him off Eric Brown pass.
Subhashish, who took the penalty, made no mistake in converting the chance past Kunal in the 46th minute.
The visitors improved their performance by several notches thereafter and enjoyed better share the ball possession but lacked the ideals inside the rival box.
Mumbai had their best chance towards the end but Pratik Prabhakar Chowdhary's fierce shot was well saved by Sporting keeper Arindam Bhattacharjee.
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