In a closely-fought match, it was likely that one goal would be the difference.
A frantic start from both sides – perhaps to do with the less-than-perfect conditions – meant there were no shots on target until the 28th minute.
The best chance fell to Tynecastle’s Kieran Ngwenya but, despite latching onto the ball quickly, opposing goalkeeper Niall McFadden was equally as sharp to smother the shot.
After two half-chances for Spartans’ impressive Reece Mason – their attacking focal point – it was Lorcan Gray, who scored what turned out to be the only goal of the game.
It was an emphatic finish into the roof of the net from Gray, who received the ball from the right ten yards out.
In the second half, Tynecastle applied the pressure in search of an equaliser and a goal looked like it was coming.
Two more chances from their dangerman Ngwenya went begging, though, and they were running out of time against a resilient Spartans defence.
The game was nearly out of sight when Spartans’ man-of-the-match candidate, Nicholas Reid, held off a tackle on the right to burst through on goal and play in Jack Crombie, but his shot was saved by Aaron Heap.
Spartans didn’t need a second goal, though, as they held on to progress to the semi-finals.
• Hutchie Vale 13s win thriller
HUTCHISON VALE 13s beat Penicuik Athletic YFC Blues 5-3 as a thrilling Lothian Buses Cup semi-final required extra time.
Mark Deya scored a double and Max McGinley grabbed one for Penicuik but it wasn’t enough in the end.
Taylor Stevens responded with a brace of his own, along with goals from Bruno Davidson, Sam Kilboy and Aiden Eddington to secure a final place for Hutchie.
There were goals galore again in the Under-14 Jewel Miners Cup semi-final as Leith Athletic beat Longniddry Villa 5-4 after extra time.
Striker Rhys McMurdo grabbed four for Leith while Strathy Mackay scored a header. Kai Bailey and Arron Darge scored a goal apiece for Villa and Byron Archibald notched two.
• Edinburgh South show their Steel
In the Under-15s Division 2, Edinburgh South CFC defeated Dunbar United Colts 3-0 with two goals from Bruno Steel and an early second-half effort courtesy of Finn Higgins.
A terrific performance from the home side saw them grab their eighth win of the season keeping them firmly in mid-table. The surprise win came against Dunbar side, who are sitting second in the league and on good form, but couldn’t come up with anything on the day.
Leith Athletic Colts ran out 3-2 winners over Hillfield Swifts in the Under-16 Division 2.
Kerr Milne and Robbie Macdonald (2) scored for Leith, who currently sit seven places behind their opponents.
Swifts’ strikes came courtesy of Marc McGuire and an own goal.