In the end, there really was no "winner" as the lines were reuinited sufficiently by the smooth ascension of Henry VIII (the issue of Henry VII, heir to the Lacastrian claim, and Elizabeth of York, last heir to the Yorkist claim). Additionally, the "Wars of the Roses" (a 19th century term) or more accurately The Cousins Wars were a series of many conflicts between the houses of Lancaster (unquestionably usurpers to the throne) and the house of York, which held the true claim to the throne.
The claim that Lancastrian descendents were the ultimate victors is supported by the fact that their ultimate successors, the Tudors were descended through John of Gaunt (founder of the House of Lancaster and third son of Edward III), rather than from Lionel of Antwerp (second son of Edward III whose great granddaughter was the mother of Richard Plantagenet, Duke of York).
Strictly speaking the first Tudor King, Henry VII's claim to the throne (besides right of conquest) is traced from John of Gaunt through his mistress (later wife, later legitimized) Katherine Swynford. His ancestor being John Beaufort, 1st Earl of Somerset. The Somerset line were always Lancastrian supporters and the 4th Earl (2nd Duke as his father had been elevated by Henry VI) was ultimately the blood enemy of Richard, Duke of York.
The 1st Earl of Somerset's granddaughter was Margaret Beaufort, who married Edmund Tudor, himself posessing strong Lancastrian ties in that his mother was Catherine of Valois, the widowed Queen consort of the great Lancastrian King Henry V.
So Henry VII was, by all manners, a direct continuation of the Lancastrian claim to the throne. He was a great-great-great grandson of Edward III through John of Gaunt, 1st Duke of Lancaster.
All of this became academic, however, when Henry VII agreed to marry Elizabeth of York (sister of the famed Princes in the Tower, daughter of Edward IV, and sole heiress to the rights, titles and styles of the Yorkist claim to the thromer) before invading and ultimately defeating Richard III at Bosworth.