Having won their second match, East Bengal went atop the table with seven points from three matches and kept their pre-quarterfinal hopes alive.
It was East Bengal coach Trevor James Morgan's previous experience in Singapore that came handy after they got a surprise 1-0 lead with a 19th-minute own goal by Anaz Hadee.
However, the Rovers were quick to respond equalising through their Serbian recruit Sead Hadzibulic in the 28th minute as both the teams were locked 1-1 at halftime.
Rovers found the target the next minute when Khairul Amri pulled one back for the home outfit.
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But the Rovers could not find another equaliser, while East Bengal completed the tally with Barisic completing his double in the 87th minute to hand them a fine away win.
Morgan, who was a coach in the S-League, had just one draw from nine previous matches. So, the win is a first for the Englishman.
The home side, who is the S-League leaders with 18 points from six matches, thus remained winless from three matches.