Ironically iv'e had to deal with 3 rat invasions in the last 3 months one at my house one at my mothers and another at my grandparents. I've found each time that one of the main causes for them to shack up is food, my mother and grandparents problem was bags of birdseed being in there garage or just inside the conservatory, my problem was i was putting to much birdseed out so the first step, would be to make sure there are no food sources available to them that will help a tremendous amount.
Secondly my grandparents used rat poison but me and my mother refuse to use it because of dogs, but the method is still similar, try and fill in the rat holes with dirt and really compact it down afterwards try and get some mesh (obviously steel is better) but anything tough will do and lay that on top of where you filled in the rat holes. After that you can lay slabs etc or something heavy just to put some weight on it. Now of course they are going to burrow out of there but if u make it difficult for them to live there and deprive them of food they may go away, you'll never stop them coming in the garden of course but they may nest somewhere else. (you can buy the poison sachets and just use a stick and stuff them down the hole as far as they'll go then fill with dirt and continue the process).
One thing that i have found a lot of people do which is the main cause of rats is put way way way to much bird seed down, only put enough so that the birds will polish it off before nightfall, and don't feel stingy or bad they dont mind Other than that there are rat traps or exterminators but Ive never been down that route so i cant comment on it.
But honestly if they are a real persistent problem you may have to bully them out with poison or a professional, try the nice approach but be prepared for a fight if it doesn't work.
Hope this helps and i hope you get the rats out of the garden !