Adolfo Miranda scored the all-important goal of the match in 49th minute.
The first half started at a fast pace and TRAU got an early free-kick in the attacking third, but Chennai's Roberto Eslava cleared the ball in the box safely.
Chennai were building patiently from the back and their new signing Katsumi Yusa was already creating problems down the left flank with some skillful and deceptive movements.
The home side soon started asserting their class as last season's joint top-scorer Pedro Manzi went very close to getting his team in front but his header went over the post.
In the dying moments of the first half, Chennai again came close to scoring. This time it was Katsumi, but his shot from close range on the near post went just wide. Despite Chennai's dominance, the first half ended with both teams locked at 0-0.
The deadlock was finally broken in the 49th minute when a miscued clearance gave Chennai's new signing Miranda an easy chance to tap it over the goal line.
Chennai could have soon doubled the lead when a brilliant run by Katsumi led to an even better cross straight at the head of the unmarked Mashoor Shereef. However, poor finishing denied him of his first I-League goal as his effort went just wide of the post.
With 15 minutes left in the match, TRAU made two changes in order to make things happen as they brought on Deepak Devrani and Abinash Ruidas. However, Chennai dominated the game till the end while TRAU somehow survived the scare.
Pedro Manzi could have closed the game once and for all in added time, but TRAU custodian Gurpreet denied him, as he dived low at the near post to tip Manzi's shot away.
( With inputs from IANS )