For a long time I used Sakura Pigma Micron pens, the .005 size. In addition to black, they also have several other colours, including an incredibly useful brown (it has reddish and yellowish elements to it, rather like the old GW Gryphonne Sepia) which can be used for other things besides eyes (I most frequently use it to do eyebrows on folks with fair hair - when I do eyebrows sometimes I paint them, and sometimes I use a pen, if they need to be super-fine, such as with my version of Red's face, it'll be the pen for sure).
I still use the Pigma Microns now, but in the last year I found the Copic Multiliner series, which is black only but has a pen about half the size of the Pigma (they sell it as '0.03' - different companies all have their different nomenclatures) and that has been INCREDIBLY HANDY, for figures with smaller eyes or anything a little more difficult.
The trick when using pens is to dab things on carefully and lightly. You still need to be able to get in close and get a good look at what you're doing, just as you would with painting - to me the real luxury isn't that the pen is tiny or anything. Really good quality brushes can paint extremely fine lines and tiny dots. The problem is that with eyes, everything needs to be placed so perfectly that you need extremely good concentration and have to be very slow and careful. I can do this vastly more easily with a pen because it won't dry out like paint on a brush will.
I should also point out that conventional painting techniques for eyes involve a multi-stage process. First you lay a slightly darker area to define the eye, then a slightly smaller area of near-white on top, then finally the pupil (well, the iris, actually). I still do the first two by conventional paint and ink washes. Sometimes pens are used to darken part of the ring around the eye to help define it (especially in cases like say, a woman expected to be wearing eyeliner or to otherwise have dark lashes, usually this is only the top of the eye and very rarely all the way around).
Which is a very roundabout way of saying pens are an incredible help, but they won't do everything. They're a very specialized tool that can help with certain specific things, but overusing them can look just as bad as not using them at all.
I have one or two figures where I have actually painted coloured irises and used the pen to actually do a pupil, but wow is that ever agonizing to try and pull off.