Technically speaking the change occurred in 1804 (prior to the election) after the 12th Amendment was ratified. This Amendment literally changed the system such that instead of the VP being the runner up, electors cast separate votes for the Preseident and Vice President.

However, in reality parties had intended Presidentital and Vice Presidential candidates in the elections of 1796 and 1800 also. In these elections it was understood by the parties (Federalists and Republicans) who was the candidate they intended to be president and VP. The intention was the electors would hold back some votes from the intended VP causing them to come in second. However, in the election of 1800, this did not happen and Burr and Jefferson received equal votes (although it was understood by the Republicans that Burr was intended to be a running mate and to be VP). The 12th amendment was passed to prevent this kind of issue arising again.

In most subsequent elections there was a clear President/VP match up. The only exception I believe would be the election of 1824, which was a very unusual election. Because the Federalists party had collapsed and could no longer realistically contend the Presidential election, the election of 1824 was a four way race among candidates of nominally the same party. In this case Calhoun was essentially the running mate of two different candidates. This lead to him winning the VP position with a majority of electoral votes despite the fact no presidential candidate had enough votes for a majority (John Quincy Adams would win the Presidency after it went to the House to decide).