Iain Hume's 23rd-minute goal led the Blasters to their important victory at the Jawaharlal Nehru International Stadium.
It saw David James' men finish fourth in the table, setting up a semi-final clash against Chennaiyin.
Chennaiyin drew 2-2 against the Delhi Dynamos, who missed out on a spot in the finals after finishing fifth.
Alessandro Del Piero's tremendous 30-yard free-kick in the 53rd minute gave the Dynamos hope of a comeback win after they fell behind 2-0 in the first half.
Hans Mulder drilled in an 88th-minute equaliser, but they were unable to find a winner and had Morten Skoubo sent off.
Atletico de Kolkata and Goa will meet in the other semi-final after playing out a 1-1 draw.
Antonio Lopez's Atletico are winless in five matches after Wednesday's result at the Salt Lake Stadium.
But the draw saw Goa finish second and Atletico third at the end of the regular season, meaning they will face off in a two-legged semi.
Edgar Marcelino's powerful 27th-minute strike had visiting Goa ahead.
Fikru Teferra's well-placed penalty in the second half secured a point for Atletico after Goa defender Bruno Pinheiro had been sent off.
Also on Wednesday, NorthEast United drew 1-1 against Mumbai City to finish bottom of the table.
