Ok, I spent about 1.5 hours figuring out how to do it, so I want to share this knowledge, maybe it will help someone.
If you can't get links like dota2://matchid=66135569 to work, you just need to save following text as a .reg file and execute it.
Windows Registry Editor Version 5.00
@="\"C:\\Program Files (x86)\\Steam\\steam.exe\" -applaunch 570 \"%1\""
UPD: by the way, it will also allow you to open these links even when your dota 2 is not running! As far as I know, usual settings just start dota2.exe, but you need to start it from steam in order to connect to servers and stuff.