Winger Gavin Sharp bagged a hat-trick of tries while centre James Elliot weighed in with two in Thurso to leave the Midlothian men in third place after their first run out in seven weeks.
Prop Craig Adam was also on the scoresheet and was later named man of the match.
Lasswade had been helped in their preparations for the long trip north by a training session with Premier One pacesetters Currie last week and president Ian Barr said: "That training session was important because it was a tough one for the boys and helped them get their match fitness back after such a long time without a fixture. The team played well against Caithness. It was a solid victory."
In the same division, Preston Lodge pushed leaders Greenock Wanderers all the way before losing out 24-19 at the weekend.
Livingston defeated Murrayfield Wanderers 23-0 in National One, while Edinburgh University lost 15-0 at home to East Kilbride and RHC Cougars went down 21-10 at Linlithgow.
Ross High beat Forrester 26-6 in National One, but Trinity Accies lost their ninth match of the campaign 16-10 at Helensburgh.
Title-chasing North Berwick and Portobello both won at home in East One. The former defeated Hawick Harlequins 43-14 and Porty got the better of Earlson 26-5. Leaders Moray House won at Inverleith in East Two.