Ian McCafferty, Michael Saunders and Kristin Forbes all voted for a rise.
Inflation is close to a four-year high of 2.9% - far above than the Bank's target rate of 2%.
Ms Forbes had been the sole vote in favour of a rise at the MPC's previous two meetings.
However, the decision of Mr Saunders and Mr McCafferty to back an increase as well surprised economists.
"The more hawkish tone of the MPC appears to reflect some concern about inflation - which has accelerated faster than it expected over recent months and is now forecast to exceed 3% this year - as well as the strength of employment that is continuing to erode slack in the labour market," Capital Economics said.
Hargreaves Lansdown senior economist Ben Brettell said it appeared that the "willingness of the MPC to 'look through' higher inflation and leave rates on hold is wearing thin, and if inflation continues to surprise we could see higher rates by the end of the summer".
Sterling, which had been trading below $1.27 before the minutes were released, surged to almost $1.28 in response, while the FTSE 100 fell more than 1%.
In the minutes of its meeting the MPC said the "driving force" behind the recent pickup in inflation had remained the depreciation of sterling associated with the UK's referendum on EU membership.
However, it added that a "number of indicators of domestically generated inflationary pressure" had also increased in recent months.
The committee said inflation could exceed 3% by the autumn and was expected to remain above the 2% target for an "extended period" as the weaker pound pushed up prices while pay growth remained "subdued".