When DSLRs are not using Live View they autofocus much faster than Mirrorless cameras. Why is that? Why can't Mirrorless cameras use the same autofocus mechanism that DSLRs use?
To answer your question directly, I think that your base assumption that Mirrorless AF is slower than DSLR AF is somewhat dated and also unfairly groups cameras that vary a great deal into a single category.
First you must understand the difference between contrast detect AF and phase detect AF. At the end of this answer, you can find more resources on this site related to that question, but I like this summary over at LensRentals.com blog best:
...contrast detection is simpler, cheaper, smaller, and theoretically more accurate than phase detection autofocus. But it is much, much slower. Camera companies are working hard to speed up contrast detection autofocus, and are making some strides, but for the near future it will remain slower.
Vast improvements have been made in the recent year or two that allow Mirrorless cameras to utilize either hybrid systems of AF or essentially the same systems as DSLRs. Early Mirrorless cameras were slow at AF because they used strict contrast-based AF and were not very refined at that. More current Mirrorless cameras, especially from some manufacturers can use phase-detection AF in live view or offer hybrid phase/contrast systems.
To not date this question too much but still give an example, take a look at the Sony A6000, which has 179 point phase detect AF. This advantage should not be overlooked. Having points across the whole sensor vs clumped together in the middle as is the case with DSLRs is a huge advantage especially in action shots. Another big advantage that you can find with mirrorless is the ability to shoot continuously with phase detect AF, saving you the time that a mirror typically has to flip up on a DSLR.
Mirrorless cameras have come a long way from the first few generations in many respects but certainly in autofocus performance. It wasn't until 2013 that phase detect AF really started to appear in Mirrorless cameras, but only a short time later they truly are challenging DSLR autofocus performance. The performance will vary across models so one cannot safely make a statement that all Mirrorless cameras perform a certain way in regards to AF.
The best mirrorless cameras match DSLRs for fast autofocus.
...the gap in autofocus performance with the best CSC cameras is closing rapidly.
[Nikon D4S AF] compared to the Panasonic GH4; they are so close I would say they are equal.
We have quite a few questions already on the site that cover this, but you may need to read multiples of them to really get a good grasp on this subject. Try these: