What I always do is look out for them in the spring and fall, when the soil is soaked (for the roots to come).
At less than a foot high, grasping the stem firmly usually does it, with a reasonable strength exerted.
At more than a foot high, I always carry a shovel, get a good grasp on the stem with a leather-palmed garden glove, then pull and dig out the root simultaneously.
I have good success with that method. But I understand larger shrubs require a lot of work to get all the root out. Keep up!

Greetings! Location is everything, this method will not work in a planting bed. In hedge rows or wilds, I lop off the stem/trunk within a foot of the ground. Using a steel spade or hatchet, I split the stump or remainder of the stem. Then, pack a handful of Halite or rock salt into the wound. As the salt dissolves, it is absorbed into the Buckthorn's root system. Dead!! Forever.

Matt, good idea to pack a poison into the wound. Though I might use Round Up Brush Killer sprayed into the split stump. It's not supposed to stay around for years like salt would.