The hosts, who are Capital Premier League campaigners, made it clear they weren't intending to be the nice guys, with James Eadie being booked after 30 seconds for a reckless challenge on Bevin.
Rovers centreback/coach Bill Robertson picked up a yellow four minutes later for taking down an energetic Facundo Barbero.
In the second half, Island Bay player-coach Martin Pereyra-Garcia was dismissed.
With their backs against the wall, the Blues needed something special and it came from the boot of Bevin who scored from a free-kick from some distance to ensure they were level, 1-1, at halftime.
In the 58th minute, Josh Stevenson slipped the ball past Bay-born goalkeeper William Stockill to put Napier ahead, 2-1.
Ryan Tinsley slotted a penalty kick in the 75th minute to make it 3-1 after a defender brought down Bevin in the box.
The English import reloaded from the penalty spot again in injury time, after Bevin drew another foul in the box, to put the host out of their misery.
Robertson was happy to have overcome the giantkillers who had knocked out the favourites Miramar Rangers in the last 16 in what seemed to be more a physical threat to their cup hopes.
"We knew it was going to be a difficult game. They had done well in the previous round so we were aware they were a capable side.
"But we controlled the majority of the game and created a greater number of chances."
The coach was prepared for a bit of rough and tumble from the outset.
"They've come to make it difficult for us and we expected that so we tried to match it.
"Yeah, it was pretty niggly at times and there were a few tackles made - not all legitimate - and a bit of pushing and shoving.
"But you expect that in cup football and if don't get that it wouldn't be a proper cup tie."
Bevin, playing in his penultimate match in blue before jetting back to the United States for studies, said the commitment of the lower-league side had caught them out.
"They wanted it more, particularly in the first 20-30 minutes, and we were lucky to go in one all. They showed far more than we had during that spell.
"But we just had to be patient and continue what we were doing.
"Even though they were a goal up they weren't really causing us a lot of problems so it was just about staying on the ball and doing what we were doing," he said, adding their quality surfaced.
Any chance of an Island Bay comeback at 3-1 down was scuppered when Pereyra-Garcia saw red.
The ref had already booked him for his reaction to a Napier foul before receiving his marching orders for a high challenge on Bevin, another player they targeted.
A packed crowd watched but the promised Yellow Fever-inspired temporary stand never appeared after the club felt the council's demands were unachievable.
