Normally, you simplest way of expressing this is to spell out the names of the months.
If this is April, then just say the following:
Things happened in February and March.
There is no way of misinterpreting that, and it's also the shortest sentence.
If you say in the last two months, then, yes, it could be misinterpreted as meaning March and April—not February and March.
It's possible you could say the following:
Things happened in the last two completed months.
Since April isn't completed yet, it would likely be assumed not to include any part of April.
However, I would still not rely on everybody understanding that interpretation. And, even if they did, it sounds odd, because it's not normally expressed in that way.
But in the case of a prize, needing to have made a purchase, and being unable to give the names of the months, this might work:
To qualify, you must have made a purchase in the previous two months.
However, I've been told in comments that some people don't find there to be any difference in meaning between previous and past in this sentence—and they would still assume it was talking about the last 61 days.
So, if you really want to make sure that the point is made, you can make it even more explicit:
To qualify, you must have made a purchase in the (last / previous) two months. Any purchase made in the current month does not count.
Or, perhaps more simply:
To qualify, you must have made a purchase in the two months prior to the current month.