New signing Dylan Dailly netted a first half double to put Rovers 2-0 up at the interval before killing the game in the second half by scoring the rebound after his initial penalty had been saved following its award when Mark O’Reilly was brought down in the box.
Scott Burnett beat Adam Brown with a late penalty – awarded for a foul – to pull one back for the visitors.
“I was delighted to win because we’ve basically been on our knees with injuries,” Rovers assistant manager Kenny Neill told the Carluke and Lanark Gazette.
"It was a struggle to get a team on the park so it’s the kind of result everybody was needing just to give us a wee lift again.
"Playing Auchinleck gives the club the chance to get a bit of money. The way things have been over the last year, we will take everything that’s coming our way.
"I dare say all the players will be looking forward to it.”
Lanark United also qualified for round two – and a home tie against East Kilbride Thistle on September 25 – with a 4-3 win at Cruden Bay with goals by Rhys McCluskey, Cammy Lawson, Jordan Conlon and Jack Hamilton.
Lanark boss Colin Slater said: “We were pleased to get the win. We were 4-1 up and it could have been a bit more comfortable but we did what we set out to achieve.
"They scored in the 88th and 89th minutes which made it more exciting than it should have been. We had another goal which was perfectly fine but the ref ruled it out.”
But Lesmahagow Juniors went out by losing 5-4 on penalties to Dundee North End after a 2-2 draw at Craighead Park. ’Gow’s goals were scored by Lewis Hill and Ross Marchant.
Forth Wanderers’ scheduled Conference A game at home to Bellshill Athletic was called off due to Bellshill gaffer Derek Wilson and seven of his players going into self isoltion after catching coronavirus.
Fixtures this Saturday with 2pm kick-offs are: Conference A – Johnstone Burgh v Forth, Lesmahagow v Girvan; Conference B – Carluke v Greenock Juniors; Conference C – St Anthony’s v Lanark.