I'm going to focus this answer on cotton, because it's been intensively researched by American academics. 

The answer to this depends at what level you are thinking about. The price of a product, and thus the profitability of producing it, is set by the interaction of supply and demand - so, taking demand as given, what is important for price-setting is the situation of what economists call "the marginal supplier", relative to the total. If the marginal supplier can reduce their production costs by use of slavery, then they will be under competitive pressures to do so. If they can't, then prices will rise slightly, up until it is profitable for them to produce, or another supplier becomes the one acting at the margin. (I hope this is clear, it's hard to explain the interaction of supply and demand without using graphs). 

To turn to history, US cotton exports continued after the civil war. The US's Federal Reserve in May 1923 published an article on cotton financing. As part of this, they discussed cotton exports, and published a table (page 567 or 33 in the pdf), showing cotton exports of 1.9 million bales in 1850 and 2.9 million bales in 1870, rising up to a high of 10.7 million bales in 1911, so clearly the end of slavery didn't render US cotton exports uncompetitive on world markets.

The end of slavery is estimated to have resulted in newly freed cotton workers reducing their labour supply by up to a third (entirely reasonably), which hit cotton production hard. The white planters responded to this (and the other impacts of losing the Civil War) to a significant extent by breaking up the large plantations and instead renting small parcels of land to the new freedman - thus sharecropping. Interestingly, cotton productivity relative to labour is estimated to have gone up during this period despite the fall in hours worked: the hours of labour required to produce a 100 lb of cotton lint was estimated at 82 in 1860 and 68 in 1870 (Craig and Weiss, 2000, page 8): cotton pickers may have been working fewer hours, but they were working more efficiently. 

However this is comparing the ante-bellum and the post-bellum periods together as a whole. An individual cotton farmer in the 1850s who didn't use slaves might well have found his (or her) cotton priced out of the market. This is not to say that every cotton farmer in the 1850s had to use slave labour to be profitable - individual farms have their own variations in terms of access to transport, soil quality, rainfall, etc. And a producer in another area entirely might well not need to use slavery at all - the US northern states were industrialising and exporting with free labour. 

As well as this, cotton is a bit of an unusual example in that a particular variety of cotton, G. hirsutum, made for very nice cotton and US slaveholders developed varieties that grew particularly well in the climate and soil conditions of the US south (Olmstead and Rhode, 2008). Adapting this variety to other primary cotton-growing regions of the world, was a much slower process. This likely contributed to US cotton exports' profitability after the end of slavery. It may be that for other crops, an individual country banning slavery would price its producers out of the world market if its 'competitors' didn't do the same. 

In short, in a counter-factual world without slavery, colonial exports would still likely be profitable. That doesn't mean that an individual producer, or a given country, could necessarily drop slavery and still be profitable if others continued to hold slaves. As the discussion about cotton indicates, details matter a lot. 

Sources

Ransom, Roger. “Economics of the Civil War”. EH.Net Encyclopedia, edited by Robert Whaples. August 24, 2001. URL http://eh.net/encyclopedia/the-economics-of-the-civil-war/

Craig and Weiss, 2000 “Hours at Work and Total Factor Productivity Growth in the Nineteenth-Century U.S. Agricuture” (chapter 1 of New Frontiers in Agricultural History, ungated copy at https://pseudoerasmus.files.wordpress.com/2015/11/hours-at-work-and-total-factor-productivity-growth-in-nineteenth-century-u-s-agriculture-copy.pdf

Olmstead, Alan L. & Rhode, Paul W., 2008. "Biological Innovation and Productivity Growth in the Antebellum Cotton Economy," The Journal of Economic History, Cambridge University Press, vol. 68(4), pages 1123-1171, December.