It's not irregular but all you can do is just to memoize such cases.
While the ст
-> т
happens in some verbs like "плести"/ "плету", "цвести"/"цвету", "мести"/"мету" for instance, a change to д
is not that rare as well, check out for instance:
- блюсти / блюду
- брести / бреду
- красть / краду
- прясть / пряду
So it's not something exotic. Here's a small explanation I borrowed from this book:

"Der Infinitiv läßt das nicht erkennen, da nach slavischen Lautgesetzen д, т vor "т" zu c wurden"
Or, in my free translation - the infinitive doesn't indicate this because, due to Slavic sound law д
and т
before т
became c
, so "вести" was initially "ведти" and what you've found are the traces of that old spelling.
This is, by the way, the case in all examples mentioned above. If you take a closer look you'll see that 'красть' was derived from "*kro-d-" stem, "блюсти" can be traced to Proto-Indo-European *bʰewdʰ-, "брести" is from " *bʰredʰ-." and "прясть" we got from "*prędti".
The scientific term for this changes is "the assibilation of dental stops". The last thing I want to add that this happens very likely in Pre-Slavic since Slavic languages share such similarities with Baltic languages.