The Royals got the better of Leeds 1-0 away from home at Elland Road in the Championship on Saturday afternoon.
An 84th-minute goal from Barrow, who can operate as a winger or as a forward, proved to be the difference between the two sides.
The 25-year-old spent the second half of the 2016-17 campaign on loan at Leeds from Premier League club Swansea City.
The Gambia international attacker joined Reading from Swansea in the summer of 2017 for a transfer fee reported by BBC Sport to be around £1.5 million.
Barrow has taken to Twitter to give his reaction to the win for Reading against Leeds.
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