Balwant scored in the 28th minute in a match the visitors played without any foreigners.
Bagan's U-22 player Subhashish Bose picked a second yellow card in the 63rd minute to be given the marching orders.
Churchill got their best chance in second half injury time when substitute Angelo Colaco's strike hit the crossbar from the edge of the box.
Bagan started better but lacked the cutting edge in their attacking. But they went ahead two minutes before the half-hour mark when Balwant powered in a header off a Pritam Kotal cross from the right flank.
The hosts went into the break leading by the solitary goal.
In the second half, Churchill fared better but failed to create any opportunity clear-cut chances, despite being a man up.
Bose was shown a second yellow card for a push on Seriton Fernandes after getting first booked in the 55th minute.
The last minutes of the match were tense for the home side as Churchill tried to wrest the advantage but Mohun Bagan managed to hold on.