Despite a good early opportunity for Tony Muir, the hosts took the lead four minutes before the break. Midfielder Davie Cameron slotted past Bryan Young after Aiden Ferris had charged down a clearance and picked out his team-mate.
BSC sealed the points after the hour. Just five minutes after coming off the bench, youngster Riccardo Corrieri glided past his man and crossed for Aiden Ferris, who slid home at the back post to make it 2-0.
Next for Welfare is their first home game of 2016, with Selkirk the visitors to Ferguson Park on Saturday.
Civil Service Strollers lost 5-0 to East Kilbride in the semi-finals of the Football Nation East of Scotland Qualifying Cup. Kilby will face Spartans in the final.