What is the significance or meaning of 筹 in 略胜一筹？
筹 (chip) was a counting tool in ancient time. It was used by mathematicians for calculating. 

運筹 was then used as a metaphor for the calculations strategists and military experts did  

Idiom: "運筹帷幄" (moving chips inside the tent) means "strategizing in headquarters" ( before go the field or not even need to be in the field)

胜一筹 literally means "one step ahead" and it evolved to mean "one up"; "bested (v)" and  "a level above" 

略胜一筹 (slightly better by one calculating step) means "slightly a step ahead" or "at a slightly higher level"

筹 is not a classifier but a noun for "level" in modern Chinese, e.g. "技高数筹" (a few levels higher in skill); "力胜一筹" (one level higher in strength)


  更胜一筹， 略胜一筹：'slightly better', 'a cut above' are correct translations