Jed started the better of the two sides and should have taken the lead through Stephen Reilly, but the big defender passed up a golden opportunity with the goal at his mercy.
Reilly did find the net shortly after, but unfortunately it was at the wrong end to put the visitors ahead.
Chris Baillie put the hosts level almost immediately, finishing off a good move involving Steven Thomas and Aaron Swailes.
Jed then fell asleep defensively, allowing the Berwickshire outfit to edge ahead.
Sean Wood levelled for Jed before half-time when he was on hand to tuck the ball home from inside the box.
Greenlaw upped their game in the second period and scored twice without reply, helped by some poor marking in the Jed defence.
Match sponsor: Gilbert Developments.