Chhangte was superb on the night while Fernandez helped seal the win for Delhi with a late penalty.
Sunday's win was, however, not enough to lift Delhi off the bottom of the table as they continue to sit at the 10th spot with a meagre seven points from 10 matches.
They have the same number of points as ninth-placed NorthEast United who are ahead due to a better goal difference.
What was expected to be a mere formality for Bengaluru FC to pick up three points and return to the top of the table turned out to be Delhi's most prized victory in this edition.
A victory that keeps them alive in the league and gives them the confidence that they can put it across any of their rivals.
Delhi started the game on the front foot, putting the visitors under pressure right from the kick off.
But the first big chance of the game came Bengaluru's way. Dimas Delgado's defence splitting pass in the 18th minute found left back Harmanjot Khabra through on goal with just Delhi goalkeeper Arnab Das Sharma to beat.
But Khabra's shot was too close to Sharma who made the save by rushing off his line.
In the second half, both sides continued to create chances but without an end product.
It looked like Delhi were again going to be unlucky on a night where they outplayed their opponents for much of the game.
Chhangte, though, had other ideas. He finally broke the deadlock after some quick passing with Pritam Kotal gave him an opening to nudge the ball home.
Bengaluru had great chance to level things up late on. In the 87th minute, John Johnson found himself unmarked in the Delhi box from a free-kick close to the corner flag. But his header went straight to Sharma who held on to the ball.
There was more drama at the end. In the dying minutes of added time, Delhi were awarded a penalty for a foul on Guyon Fernandez by Subhasish Bose, who was shown a direct red card.
Fernadez got up and converted the spot kick to give Delhi a famous victory.
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