A shufti at the list of Senate roll call votes (there haven't been too many so far!) shows that the piece was talking about the cloture vote on the bipartisan substitute SA 3.

(Heaven forfend that the Times would spare us kibitzers a needless paperchase!)

The vote page is clear: a 3/5 majority (duly chosen and sworn) was required. So - where did the Times's 65 come from?

Must have been applying the cloture rule applicable to changes in the Senate rules - 2/3 of those present and voting. Since there were 3 non-voters, that would make 65 senators.

But - S 1 amends both Senate Rules and Federal laws. Bugger!

The wording of Rule 22 is ambiguous:

...if that question shall be decided in the affirmative by three-fifths of the Senators duly chosen and sworn -- except on a measure or motion to amend the Senate rules, in which case the necessary affirmative vote shall be two-thirds of the Senators present and voting --...

I'd say that the 2/3 exception only applies where the measure or motion is exclusively to amend the rules.

But - technically what the SA 3 cloture vote was on was a motion to amend a bill that would, if enacted, serve (in part) to amend Senate Rules. (One level removed
from the measure or motion to amend referred to by Rule 22.)
I've had a quick squint at Riddick's but come up empty: on page 326, the entry under Rule Change just says

Cloture, by a two-thirds vote, may be invoked on proposed amendments to change the rules.

Hmm...