The Old Anniesland side are still unbeaten in all competitions this term after the superb 21-7 victory against Peter Wright’s Borderers.
The giant killers will now await the winners of the postponed Stirling County-Ayr tie in the quarter-finals.
Selkirk, who had defeated Musselburgh in the previous round, went ahead 7-0 in the first half and that was the way it stayed until the break, however GHK came out flying after the interval and scored three converted tries of their own to progress.
Mike Borthwick, Danny Campbell and Marc Taylor were the try scorers for GHK, while Mike Martin was on form with the boot.
The result adds to Selkirk’s frustrating season following relegation from the BT Premiership.
Elsewhere in round three Jed-Forest pushed Heriot’s before the Edinburgh side came through 20-19 at Riverside Park.
A Fraser Harkness try had put the Borderers 19-12 up, but a penalty from John Semple and then an unconverted try saw Heriot’s, who won the event in 2013-14, through to the last eight.
Two early tries from skipper Ross Weston saw Currie on the way to a comfortable 31-0 win over Watsonians at Malleny Park.
That double saw them 12-0 up at the interval and then Richard Snedden, John Cox and Thomas Gordon crossed in the second 40 minutes.
The Hawick-Melrose and Stirling County-Ayr round three matches were rained off.
In the delayed round two ties there were wins for GHA, Cartha Queen’s Park and Dundee HSFP.
GHA went to Hartreemill and defeated Biggar 26-18 while Cartha had to battle hard to win 25-20 at Aberdeenshire.
Dundee HSFP will play Marr in round three after seeing off Falkirk 27-19.