According to this answer, only a simple majority is needed to change the Senate's rules not a supermajority.
Ordinarily, a point of order compels the Senate to follow its rules and precedents; however, the Senate may choose to vote down the point of order. when this occurs, a new precedent is established [...]
Before this, the previous [change to the cloture rule occurred in 1975]Broken link removed, when the Senate reduced the requirement for cloture on all matters except future rules changes:
Before 1975, two-thirds of the Senators present and voting (a quorum being present) was required for cloture on all matters. In early 1975, at the beginning of the 94th Congress, Senators sought to amend the rule to make it somewhat easier to invoke cloture. However, some Senators feared that if this effort succeeded, that would only make it easier to amend the rule again, making cloture still easier to invoke. As a compromise, the Senate agreed to move from two-thirds of the Senators duly chose and sworn (normally, and at a maximum, 60 votes) on all matters except future rules changes, including changes in the cloture rule itself.
How many senators approved the amendment to the cloture rule in 1975? (i.e. was it 2/3rds, 3/5ths, a simple majority)
See: Changing the Senate Cloture Rule at the Start of a New Congress for more recent information, including reference to the change in 1975.