Well, bitcoin confirmation time is around 10 minute. In 10 minute time range, bitcoin network collect many transactions and processing in 1 block.
Maybe your transaction was after 9 minute of the last bitcoin block, so you need only 1 minute to be processed. But maybe im wrong because this is only my thought.
It's worth noting that the time between blocks is currently around 10-15 minutes
on average, but this is highly variable.
You can look at the most recent blocks mined here:
https://www.blockchain.com/btc/blocks. If you look at yesterday (08 Oct 2018), you can see that between blocks
544839,
544840, and
544841, there was only 48 seconds - 17 seconds between the first two blocks, and 31 seconds between the next two. However, you can also see that between blocks
544893 and
544894, there was a gap of almost an hour.
So while you are correct that OP's transactions were probably just luckily timed and were included in a block quickly after being broadcast, the 9 minute/1 minute example you gave isn't necessarily true.