Ulster led 10-6 at the break but a great second quarter from the visitors saw Scotland international Chris Cusiter, sin-binned in the opening 40 minutes, grab a converted try to give Borders a 13-10 lead on the hour mark.
However, Feather's 64th-minute try, when he was driven over the line from a lineout, also kept the Irish province on course for a top-eight finish which will see them qualify for the Celtic Cup.
Centre Charlie Hore opened the scoring for The Borders when he kicked a penalty but Ulster's David Humphreys soon levelled the scores after scrum-half Chris Cusiter was sin-binned for interfering with the ball.
Ulster failed to capitalise on the advantage with Humphreys, having an off-day with the boot, missing two more penalty opportunities.
Even when Borders were reduced to playing for 20 minutes with just 14 men; Bruce Douglas following Cusiter into the sin-bin, the Irishmen couldn't stamp their authority on the game.
In fact Borders took the lead when Alistair Warnock slotted a 40-metre drop goal, his 5th of the season, to give them a 6-3 lead when they were back to full strength.
Ulster responded with the try from Ward. A cross kick from Humphreys was collected by Tommy Bowe, Steinmetz maintained the pressure by returning another cross-kick where Ward pounced to score.
Steinmetz then superbly converted to give Ulster a 10-6 lead at the break and despite Borders' spirited second half display they weren't able to score their first ever away victory in the league.
|Ulster Rugby Score Card|
|The Borders Score Card|