The idea that the 'victory' of the US/NATO and the 'defeat' of the USSR/Warsaw Pact at the end of the Cold War in essence represented the 'victory' of capitalist/free-market liberal democracy and the 'defeat' of communism is a pretty common trope in political theory post-1989.

Hitchens was referring to the form of that argument put forward by Francis Fukuyama in a 1989 essay, The End of History?, and subsequently expanded upon in his book The End of History and the Last Man. Fukuyama's thesis was that the end of the Cold War not only represented the defeat of communism, but the triumph of capitalist liberal democracy as the pinnacle of human socio-economic/political evolution:

What we may be witnessing is not just the end of the Cold War, or the passing of a particular period of postwar history, but the end of history as such: that is, the end point of mankind's ideological evolution and the universalization of Western liberal democracy as the final form of human government.

The Fukuyama argument is essentially a counter-argument to Marxist theory, which specifies socialism as the inevitable end-point of human political development. He was a prominent figure in the late '80s/early '90s neoconservative revival, though he eventually broke with the George W. Bush-era neocon establishment.

Hitchens' own political identity shifted pretty dramatically over the course of his life. He was for much of his life a Marxist/socialist; it wasn't until post-9/11 and deep into the era of the 'Global War on Terror' that he became more widely identified with the neoconservative or even neoliberal movements (depending on who you ask; he was nothing if not divisive.) So, for example, here is a pretty damning Hitchens critique of The End of History — I'm not entirely clear on its provenance, but it sounds like it dates from the early '90s.

So, in A War to Be Proud Of in 2006 — by which point Hitchens' ideological nemesis is Islamic fundamentalism, not liberalism — he's describing Fukuyama's thesis, but he's not agreeing with it. The full context of the quote (emphasis mine):

I once tried to calculate how long the post-Cold War liberal Utopia had actually lasted. Whether you chose to date its inception from the fall of the Berlin Wall in November 1989, or the death of Nicolae Ceausescu in late December of the same year, or the release of Nelson Mandela from prison, or the referendum defeat suffered by Augusto Pinochet (or indeed from the publication of Francis Fukuyama's book about the "end of history" and the unarguable triumph of market liberal pluralism), it was an epoch that in retrospect was over before it began. By the middle of 1990, Saddam Hussein had abolished Kuwait and Slobodan Milosevic was attempting to erase the identity and the existence of Bosnia. It turned out that we had not by any means escaped the reach of atavistic, aggressive, expansionist, and totalitarian ideology.

The crucial bit there is his use of the term 'unarguable', which seems to imply agreement. I'm not so sure — I think he's allowing that the balance of geopolitical power had shifted unquestionably in favour of liberal democracies, but he saw it as an unfinished war; one in which theocrats and dictators, not communism, were the enemy. From the conclusion of his essay (again, emphasis mine):

The great point about Blair's 1999 speech was that it asserted the obvious. Coexistence with aggressive regimes or expansionist, theocratic, and totalitarian ideologies is not in fact possible. One should welcome this conclusion for the additional reason that such coexistence is not desirable, either.

edit: redacted