I assume you are meaning Southern Rhodesia or what would become Zimbabwe, rather than the Rhodesia of the late-19th century which included what is now Zambia? If that is the case, I'll first address the few key reasons why Southern Rhodesia was seen as being economically more prosperous than many surrounding states, before moving onto the inherent characteristics that meant it could not have survived the tumultuous period of decolonization that spanned the post-WW2 era. If you are interested here is another question that I answered previously to do with the economic collapse of Zimbabwe during the early-2000s.
Economic Foundations
Southern Rhodesia was a chartered company territory until 1923. It was founded by Cecil Rhodes' British South African Company in order to gain access to the mineral rights of Mashonaland and Matabeleland, the two principal areas of Southern Rhodesia. From 1890 and the Pioneer Column to about 1910 (i.e. the first twenty years of the colony's existence) the primary focus of the Company administration was mining. Ian Phimister's An Economic and Social History of Zimbabwe contends that the preference given to mining in terms of land access rights, labour management policies and economic incentives ensured that the mining of gold, iron and diamonds in the region substantially profited the BSAC (as owners of the territory). After 1923 and the decision to shift away from company rule to British rule, the profits that came from these mining operations were complimented by the rapid and very profitable growth in cash-yield crops including tobacco and maize. Agriculture which had been dominated by small African farmers until the mid-1910s (in 1904 90% of agricultural produce came from African farms) was now quickly replaced by large (both in terms of land and labour) white owned agriculture. Continued immigration throughout the interwar period saw an economic growth that exceeded that of other comparable colonial possessions in Southern Africa, such as Mozambique, Angola, Namibia, and Botswana, as well as a developed infrastructure, industry (primary, secondary, and tertiary) and economic market.
However, socially and politically the colony remained under minority rule (a small white population of between 200,000 and 400,000 ruled an African population of between 4 and 6 million) and consequently when Britain began to pursue a policy of decolonization and majority rule, Rhodesia could not continue in the form it was and its economic superiority was heavily dependent upon the minority rule system as well as its international ties with the British empire/commonwealth.
Decolonization and the Economic Changes
Despite growing international economic and social pressures because of their resistance and reluctance to decolonize and permit majority rule (for a variety of reasons including belief in danger to white lives, a fear of an economic collapse, and a common belief that African government would see the state itself suffer), the Rhodesian economy continued to grow during the 1950s. A large number of immigrants from Britain and other commonwealth nations immediately after WW2 helped boost the development of industry, including steel works and similar, but Rhodesia remained economically vulnerable for two reasons; i) it had no oil reserves of any significant kind and ii) it had no coastline meaning it was dependent on neighbouring states to both export and import goods.
Tobacco remained a huge part of Rhodesia's economic wealth during the 1950s and 1960s. When the Rhodesian Front government of Ian Smith declared independence in November 1965, the response by Britain (technically still the colony's government) was based around utilising these economic problems of no oil and no coastline to bring Rhodesia to its knees. Economic sanctions were fierce and as the rest of the Imperial powers were also decolonizing, Rhodesia's allies (Portuguse territories in Mozambique and Angola) fell away. Rhodesia's only significant ally was Apartheid South Africa who saw it in their own interests to maintain a friendly white minority government to their north.
As well as economic sanctions, from the early 1960s Rhodesian found itself embroiled in a large-scale guerrilla 'Bush War' against African nationalist forces of ZANLA and ZIPRA, who sought to overthrow the white minority government and create an independent African state like had happened in Zambia, Malawi, Angola, and Mozambique. This essentially strangled economic growth as white emigrated in large numbers from Rhodesia, and businesses found it difficult to continue exporting or supporting Rhodesian agriculture due to the economic sanctions. Rhodesia depended heavily on South Africa for its exports and imports and growing pressure on South Africa to end apartheid furthered the difficulties faced by the Rhodesian government. With continuing white emigration, a costly war (in terms of money, lives, and time) and an economy strangled by sanctions, by 1979 the Rhodesian government was willing to negotiate a transition to majority rule, first in the March 1979 Internal Settlement, then the 1980 Lancaster House agreements, including the first democratic elections which saw Mugabe and ZANU-PF voted into government.
Was Rhodesia really more prosperous?
By the late 1960s, Rhodesia therefore was beginning to lose its economic prosperity in relation to other African states. Botswana for example struck diamonds in the late 1960s, and this discovery of significant minerals permitted the newly independent state to control the access, development and profits that came from the diamonds in a way which other colonial or ex-colonial states couldn't. Angola and Mozambique, both Portuguese colonial possessions and as a result subject to the Portuguese policy of captive markets for much of the twentieth century, were crippled and remain to this day economically weak. Zambia had strong international support due its anti-Communist stance and western-friendly attitude, plus large reserves of copper were invaluable economic exports. South Africa was arguably more prosperous than Rhodesia but it was also a measure of prosperity based on a small white population ruling a large African population, most of whom lived below the poverty line.
There is a popular conception of Rhodesia as being the Bread-Basket of Africa until the African took control and everything fell apart. This view is propogated and regurgitated by white Rhodesians, Rhodesian sympathisers, and white South Africans as being gospel (my own parents are guilty of holding such views). However, the reality was that Rhodesia's economic strength was hampered and in decline by the 1970s because of the economic sanctions, the civil war, and the shift away from primary industries to other paradigms of the international economy. It must also be pointed out that the policies of Mugabe in the early 1980s did little to help the transition from minority to majority rule. Despite the continued promises during the war of the creation of a socialist state and an increase in the living standards for all Zimbabwean citizens, Mugabe was quick to side with the 'Imperial Capitalists' of tobacco and minerals at the sacrifice of his own people. The answer I provided at the beginning has a section entitled 'Failure to Adapt' which discusses these problems in more depth.
And that should hopefully answer your questions. Ian Phimister's book that I reference early on in the answer is a great approach to the early economic history of Rhodesia up to the end of ww2. Otherwise Suzanne Dansereau's Zimbabwe: the political economy of decline (Uppsala, 2005) looks at the economic collapse in more detail. For the end of Rhodesia as a minority state, W.H. Morris-Jones has recently written a book called From Rhodesia to Zimbabwe: Behind and Beyond Lancaster House (2013) which is a very good look at both the factors leading up to and the actualities of the end of white minority rule in Rhodesia.
Any questions just ask!