The visitors were on top for most of the game and two tries in the final ten minutes by Matthew Hooks and Vincent Wright sealed the points.
Gregor Hunter was on target with the boot, converting all four tries and kicking a penalty.
Currie dominated the first 20 minutes and it was no surprise that they pulled 14 points ahead. Melrose were denied possession and Ben Robbins raced through for the opening try in just six minutes and Hunter converted.
The one-way traffic continued as Currie controlled up front and their backs threatened with every attack. Chris Davies found space to nip in for the second try and again Hunter converted.
Currie’s defence held out fine as Melrose at last took play inside the visitors’ 22, but lost ground when they were penalised for a high tackle.
But just before half-time, Michael Vernel was yellow-carded and three minutes later Melrose made the extra man count. Neil Irvine-Hess was at the bottom of a forwards drive to score and Craig Jackson added the conversion to narrow the gap to seven points at the interval.
Currie resumed on the offensive after the break and Robbins was tackled into touch at the corner flag before Hunter kicked a penalty to increase their lead.
Jackson again cut the deficit with a penalty awarded for a high tackle.
It became a game of ping-pong as the ball was kicked downfield in either direction, before Hooks and then Wright scored a quick double and Hunter kicked the goal points.