Underhill was almost a one-way street in the first half as Forest Green Rovers lacked any invention and were on the back foot almost from the kick-off.
They improved in the second half, but not enough to threaten the Barnet goal.
Just before Barnet took the lead, Frazer Toms crossed from the left and the goalkeeper flapped and conceded a corner. Gower took it and when the ball bounced back to him he drilled a low ball goalwards, only for defender Lee Russell to scoop the ball to safety.
The respite was short-lived. Within a minute Barnet went ahead, Ismail Yakabu won a ball and fed Gower and he drove his shot hard and low into the corner of the net.
More frantic activity from the home side reduced Rovers to just a fruitless corner and free-kick by the half-time break. Barnet continued their pressure but couldn't find the back of the net.
In October the two sides had shared eight goals at The Lawn and although Barnet looked like they might equal that score Rovers most certainly didn't.
The second half Rovers did improve, but their midfield were over-ambitious with long balls over the back line. The forwards were stranded and stood no chance of taking any advantage.
Barnet might well have gone further ahead when substitute Wayne Purser struck a post, but they left it late to ensure that Rovers didn't sneak a late and unmerited equaliser.
Gower, who worked his socks off all afternoon, crossed from the right and lurking on the far post was Junior Agogo. The young striker killed the ball and executed a volley of class into the roof of the net.
Forest did wake up and in the very last minute got their first shot on target. Alan McLoughlin hitting a hopeful shot that Danny Naisbitt had no trouble in saving.
At the other end the Forest Green Rovers keeper Justin Shuttlewood belied his two and a half years out of Conference football. Signed as a stop-gap for last week's first round FA Cup game with Exeter, the keeper showed his worth.
On four occasions he came out and denied Agogo any kind of chance