The Haddington captain repeated his try-scoring exploits from the opening day draw against the Fifers as Bryan Craig's men made it three wins on the bounce.
There was also a score for second-row Harry Hamilton on his 1st XV debut, coming off the back of another impressive display from the Reds pack.
Craig admitted he was delighted with the performance. He said: "That was the best we've played all season, especially in the second half.
"For much of the second half, we controlled the ball and that meant we controlled the match.
"We made a lot of changes to the squad this week and the boys really fronted up. The commitment and decision-making was superb."
Craig would have been delighted as debutante Hamilton rounded off strong work from the Haddington pack by crashing over early on.
Mike Legget also rounded off a superbly-worked team move to score out wide, while Cockburn powered through the Fifers' attempted tackles as the hosts made amends for their below-par opening day performance.
Nick Bell was again on good form from the kicking tee, adding two conversions and a penalty to secure his place as the club's top points scorer this campaign, and more importantly, the match.
Craig was delighted with another impressive performance from the home side, with number eight Callum Champion and winger Ewan Chalmers being singled out for praise.
Chalmers was perhaps unlucky not to be on the end of an overlap, with the winger just failing to connect with Ben Laing's pass as the ball drifted into touch.
That was one of the few chances the hosts didn't take as they secured another victory at Neilson Park.
The performance came in for strong praise, led by the club's president, Peter Brownlee. He said: "I really felt like we came of age today.
"Kirkcaldy might be disappointed not to have won here, but I think they're lucky to go away with a bonus point, because we were superb today."