In a rain affected match, Oman posted a mammoth 8/244 with Adnan Ilyas (74) and Jatinder Singh (62) scoring the bulk of the runs.
Norman Vanua was the standout bowler for the Barras, taking 4/35 off his eight overs.
Under the Duckworth and Lewis system, the Barras had to chase 252 for victory in 36 overs.
The Barras got off to a brisk start with Tony Ura (33) and Vani Vagi Morea (38) putting on 48 runs for the first wicket.
Mahuru Dai 47 and Sese Bau 35 fired in with more runs to keep the match in the balance.
Going into the last over with two wickets remaining, the Barras needed five runs for victory, with Dai and John Boge Reva at the crease.
Reva was cool, calm and collected as he smashed a boundary off the second last ball to give his side the win.