Centre-back Elphick was withdrawn just before half-time after a two-footed tackle from Ryan Lowe which saw the Dons striker sent off.
In his absence, Arter wrapped up the points with a third goal from 25 yards with six minutes left and was quick to pay tribute to their skipper.
"On behalf of the team, I would like to dedicate that win to Tommy because he is on crutches," Arter told the Daily Echo.
"He has been a great captain for us and my goal was for Tommy.
"If I had missed, the gaffer probably would have had the hump with me! Luckily for me, I caught it right, it went in and it was one of the more pleasing goals I have scored."