The game ended with Kumar pulling Nathan Lyon's final delivery for four.
"I thought we were going to get there but India hung on well," Smith said.
Smith was named man of the match and series, having scored 769 runs to set a new mark for most runs in a test series between the two nations.
"I've enjoyed every minute of it, batting out there and captaining these boys," he said.
The 25-year-old snaffled a spectacular one-handed catch at slip to help reduce India to 160-2 at tea yesterday.
A draw remained the most likely outcome at that point, but both Smith and India captain Virat Kohli were keen to force a result.
Kohli, unbeaten on 26 at the start of the final session, was looking at an equation of 188 runs at 5.7 an over.
Smith needed a wicket every 25 balls, but India blinked first.
Realistically, their hopes of victory were with Kohli and Murali Vijay. The pair had counter-punched brilliantly in the first test at the Adelaide Oval, where India threatened to pull off a remarkable first-test win after being asked to chase 364 runs on day five.
This time around, they put on a 74-run stand. It ended in the fourth over after tea, when Hazlewood started Australia's late push for victory.
Hazlewood, who finished with figures of 2-31 from 17 overs to enhance his status as Australia's new metronome, had Vijay caught behind for 80.
Kohli went six overs later, Starc inducing an edge that Shane Watson did well to snap up in the slips cordon.
Starc, recalled for the 11th time in his stop-start test career, struck in his following over to ensure Suresh Raina bagged a pair.
Wriddhiman Saha was victim of a skidding ball from Lyon that kept low and struck his pads, and Hazlewood trapped Ravichandran Ashwin lbw.
But the collapse ended there.
Lyon was the leading wicket-taker in the series with 23 scalps.
-AAP
