The LDP successfully stayed in power for such an extended period in part due to the creation of what is commonly referred to as the Iron Triangle. This is more towards the economic side of how they stayed in power and less of their exact electoral tactics which unfortunately I am not an expert on so I hope someone else will be able to give an answer on that front.

The Iron Triangle was essentially a political system which comprised of the LDP which controlled the Diet (Japanese Parliment), Keiretsus which is best described in English as a conglomeration of Japanese companies which shared interest and production lines and lastly the Governmental Bureaucracy such as the Ministry of International Trade and Industry (MITI). While this is not entirely legal, especially by western standards since it involves what would essentially be cronyism from a certain point of view, it created a large degree of political stability as this triumvirate effectively keep LDP in power and the Japanese economy booming. The Keiretsus would inform MITI and other government bureaucratic apparatus on the requirements they needed to remain export competitive such as additional loans, changes in export policy or interest rates etc. MITI as the technocrats would then draft legislation or policies that would meet these needs and then pass it on to the politicians in the LDP. The LDP using their overwhelming majority in the Diet would then provide political backing for major government actions such as loans and legitimacy for the draft policies by then passing them into law for the Keiretsus. In return, the Keiretsus provided extensive funding for the LDP elections and reserved special positions for retired bureaucrats and politicians within their companies.

This in part contributed to the Japanese post-war economic miracle making the Japanese economy boom. This economic prosperity in part kept the LDP in power as the populace was satisfied with their political leadership. The Keiretsus sponsoring the LDP during elections also ensured that they stayed in power as they were able to project a strong presence to the electorate due to the extensive amount of resources they had. This, of course, is cronyism to some extent but it should be pointed out that the initial relationship in the 1960s did not amount to any illegal or expressly corrupt actions being taken by the government and served mainly to expedite policy and actions that would be seen as reasonable. This eventually soured with the LDP's failure to contain the housing bubble which burst in 1992 as well as the banking crisis which was caused by the Iron Triangle itself as loans to Keiretsus were not closely monitored. Some Keiretsus themselves around this time also started to devolve into Zombie businesses due to the problems with the Japanese business model where firing someone was considered taboo. This led to steady unprofitability which weakened them and even forced some of them to close reducing the support within the Iron Triangle. The sharp down turn in the Japanese economy in the Lost Decade (90s) and the failing Keiretsus that reduced the support base the LDP could draw upon eventually contributed to them losing the 1993 election which ended their 50 years of dominance.

In conclusion, their political dominance was one also based on their economic success which provided them with a solid electoral base and the support of big business which contributed to their campaign. This is not uncommon in other successful Asian economies such as Korea and Singapore which also have similar arrangements.

Sources:

Japan, the system that soured, the rise and fall of the Japanese Economic Miracle - Richard Katz

National University of Singapore online resource of the Japanese Iron Triangle