She outplayed her opponent over the first nine and was four up at the turn but MacGarvie fought back to square the match at 15, and went one up with a birdie at 16.
Molly rolled in a lovely putt on 18 to square the match and take it into extra holes.
At 19 she fired in a lovely approach, with her birdie putt just slipping past. Meantime MacGarvie's approach shot was just short of the green and she failed to get up and down to extend the match.
In Thursday morning's semi-finals, Molly will play the No 2 seed, Chloe Goadby (St Regulus) who won her quarter-final by 2 and 1 against Claire Robertson while McWilliam was a 2 and 1 winner against Jillian Farrell (Cardross).
Justine Fulton beat her first round opponent but lost in the second round to Eilidh Crawford. Katie was unfortunate to come up against a girl with a 32 handicsp who was playing to beneath her handicao and to whom she had to give 26 strokes. Katie lost 4 and 3.
RESULTS
SCRATCH CHAMPIONSHIP
FIRST ROUND
Fiona Liddell (Schloss Vornhoz) bt Mirren Fraser (Powfoot) 4 and 3.
Jennifer Allan (Glenbervie) bt Emma Hale (Troon Ladies) 3 and 1
Jillian Farrell (Cardross) bt Kirsty Brodie (Strathmore) 4 and 3.
Shannon McWilliam (Aboyne) bt Louise Duncan (West Kilbride) 2 and 1
Hazel MacGarvie (Troon Ladies) bt Emma Morrison (Craigmillar Park) 3 and 1.
Molly Richmond (Strathaven) bt Megan Robb (Petercultrer) 3 and 2
Claire Robertson (Carnoustie Ladies) bt Joanne Free (Gullane) 3 and 1
Chloe Goadby ( St Regulus) bt Jasmine Mackintosh (Hazlehead) 3 and 2.
QUARTER-FINALS
Liddell bt Allan 4 and 3.
McWilliam bt Farrell 2 and 1.
Richmond bt MacGarvie at 19th
Goadby bt Robertson 2 and 1.
ANSLEY REID SALVER (Handicap)
FIRST ROUND
Chloe Henderson (Deeside) bt Jennifer Rankine (Haggs Castle) 8 and 7.
Ailsa Brannock (Castle Douglas) bt Katie Scott (Strathaven) 4 and 3.
Hannah Darling (Broomieknowe) bt Lori Macdonald (Cardross) 8 abd 7
Porscha Wilson (Burntisland) bt Emily Rigg (Craigmillar Park) 5 and 3.
Lauren Gilbert (Largs) bt Katie Malcolm (Dumfries and Co) at 19th
Megan Ashley (Strathmore) bt Shannon Pook (Auchterarder) 5 and 3.
Eilidh Crawford (Kingsfield) bt Carys Irvine (Craigielaw) 2 and 1.
Justine Fulton (Hayston) bt Jodie Taylor (Downfield) 2 holes.
QUARTER-FINALS
Brannock bt Henderson 1 hole.
Darling bt Porscha at 21st
Ashley bt Gilbert 2 holes
Crawford bt Fulton 2 holes.