Also, the CFC defence held firm despite pressure mounted by the Pune team in the dying minutes. The team had missed crucial points in the last two home games against Kerala Blasters and Delhi Dynamos by conceding goals near the end.
The former ISL champion went to the top of the standings on 20 points with the win, their sixth of the season.
Chennaiyin FC continued their dominance of the Pune outfit, having never lost in their seven previous encounters, winning five and drawing two. They won the away encounter early in December (1-0).
The introduction of Augusto did the trick for the home side. His immaculate through pass found Nelson on the right. The winger cut his man before firing his shot past the Pune 'keeper.
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The second half too saw the two teams create chances but the marksmen were not able to find success and the goalkeepers rose to the challenge.
The first half saw the teams come up with some chances with the visitors seeming to have the better of the exchanges. It was the home side which had the best chance with a penalty, which wasn't converted.
Pune City's Adil Khan took down Gregory Nelson with a heavy shove from behind with his shoulder, which prompted the referee to point to the spot.
Mihelic, who stepped up to take it, came up with a shot towards the right side of the 'keeper. Pune goalie Vishal Kaith anticipated it and dived to make a superb save.
Emiliano Alfaro looked in inspired form and created a few chances in the first half, but they didn't result in goals. Also, Chennaiyin goal Karamjit Singh did a good job in keeping the Pune strikers at bay.
Both sides created a fair number of chances in the first but their finishing let them down.