Early efforts from Sloan and Conor McKendry for Larne were comfortably dealt with by Lewis Ridd while for Cliftonville Jonny Addis flashed a left-foot shot wide of the post.
The hosts came close to breaking the deadlock when Levi Ives' powerful shot cannoned off the left-hand post and Sloan's follow-up attempt was saved by Ridd.
Then Curran headed into the net from Rory Hale's cross before half-time but the effort was disallowed.
The visitors' disappointment was compounded in the first minute of the second half as Conlan's own goal, as he was under pressure from Sloan, after good work by Ives and Thomson down the left, saw Larne go in front.
McKendry could have doubled the lead when he rifled over from close range and then Hale spurned a glorious chance to equalise as he skewed a shot wide of the post.
Aaron Donnelly and Benji Magee were involved in the build-up as Sloan drove home the second to ensure the three points stayed in east Antrim.