RMI came close to coming back from a two-goal deficit.
Neil Carroll's first ever goal for Leigh pulled Steve Bleasdale's men back to 2-1 and that was how it stayed.
Barrow had taken the lead on 37 minutes through Matt Henney and they doubled their advantage just before the hour when ex-Morecambe striker Jason Walker nudged the ball home.
Carroll's strike gave Leigh hope, but they were left to rue their missed chances.
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