'Their' is technically correct because 'winning' is a gerund and functions as a noun in that sentence, requiring the possessive form however many native speakers use either form. This site gives a more detailed explanation: http://grammartips.homestead.com/possessivewithgerund.html

"them" is corrcet.

As Susan612 has explained, technically, the correct form is 'their'. You should use a possessive adjective because 'winning' is a gerund. Gerunds function the same way as nouns, so you should say 'their winning', in the same way as you would say 'their' + any other noun. However, in informal speech, the majority of native speakers say 'them'. Strictly speaking, 'them' is incorrect, but this usage is now so widespread that it is almost acceptable.

them