I tried to find out when the first digital transaction using computers happened, but could not find a specific date. By digital transaction I mean a money transfer between two banks (I assume it was between banks at the beginning) involving an electronic mean, not banknotes, coins, notes of hand or similar.
According to Wikipedia, the very first electronic (wire) transfer was done using the telegraph in 1872 (or shortly before):
The first widely used service for wire transfers was launched by Western Union in 1872 on its existing telegraph network.
However, I cannot find any information about using a computer for a similar purpose.