The question of whether or not the development capitalism is tied to the Enlightenment is perhaps best answered by considering Adam Smith's The Wealth of Nations. It's also perhaps worth remembering that Smith considered himself not an economist, but a moral philosopher.
Smith's book is largely a refutation of the mercantilist idea that a nation's wealth is determined by its supply of capital. Instead Smith argues that a nation's wealth is determined by its "annual labor" and argues against the sort of protectionist tariffs that mercantilist theories call for. He explains how industry can be most rationally organized and how this can lead to a 'more opulent' society through increased production.
In a sense, Smith is explaining how capitalism works, but he's also dealing with laborers and how they work. By making labor the measure of a society, he is arguing that the welfare of every worker in the society is important in ensuring that the society functions. What he calls the 'progressive state' is best for the laboring poor, and thus most wealthy. He wants the laborers, who have been unappreciated by mercantilist emphasis on farmers, to be revalued. He argues that, by improving the lot of the poor, and liberalizing laws, a state becomes wealthy. He is trying to rationalize labor and explain how it might operate better, but he's also representative of the liberalizing mood of his age. The Russian Tsar's drawn out decision to emancipate the serfs was directly inspired by Smith's argument that a capitalist economy needed a free and mobile labor force, and it's worth remembering that Smith was considered a radical in his day.
Smith shares the Enlightenment focus on progress. Historically, he agrees with Rousseau that private property was the start of civil society. (766) But, he also sees states as progressing, and believes that certain practices can retard that progress. His 'progressive state' will abandon the outdated practices of slavery and primogeniture, and mercantilist tariffs that slow progress. It's important to note though that Smith is much more ambivalent about capitalism than he is made out to be. He is clear about the need of societies to increase, and cease to violate "that natural liberty which it is the proper business of law, not to infringe, but to support." (353) For Smith, individualism and liberalism facilitate capitalism. But, Smith is not the cheerleader for Capital that he has been made out to be. Prefiguring later critics, he warns that the division of labor "destroys intellectual, social, and martial virtues," and suggests that the state remedy this by rigorous public education. (839) It's interesting to compare this to Marx's claim, nearly a century later, that capitalism will break all of the old communal bonds. Given how they've been remembered, it's ironic to consider that, in terms of culture, both Marx and Smith were conservatives.
He also shares the Enlightenment fascination with human nature. He explains how the natural 'propensity' towards self-interest leads to the division of labor. Smith argues, famously, that this self-interest leads every man to look for the work that is most advantageous to himself, but that he also works, without knowing it, to promote the general interest. This is where he makes his sole reference to the 'invisible hand'. Throughout the book though Smith argues that individualism, unrestrained, will work to further society. There is again the argument that natural laws and human institutions can be reconciled with one another.
Surprisingly, for some readers, Smith is also opposed to corporations, monopolies, and most private interest groups. He believes that corporations, of any sort, corrupt the working man. (149) I'm not sure it's a far stretch to compare this to the Enlightenment case against 'factionalism'. Smith writes: "The sophistry of merchants inspired by the spirit of monopoly has confounded the common-sense of mankind." (527) In a sense, I think we could compare Rousseau's ideal of free individuals, free of factions, whose free wills added together embody the General Will to Smith's idea of liberated laborers, free of corporations, whose work added together embodies the National Wealth, and whose self-interest furthers the general interest.