This one may or may fit the post-apocalyptic criteria (I think there was an apocalypse, but the books are set on a planet that isn’t earth that was settled by people fleeing from the problems on earth), but I read them recently and really enjoyed them - , “The Knife of Never Letting Go: Chaos Walking: Book One.” The main characters are a couple of teenagers who encounter all kinds of problems and adventures on their planet.
FYI - reluctant readers may have trouble with the way the Chaos Walking series is written - with rambling sentences & some slang/written accents. Use the Look Inside feature at Amazon to take a peek. If you can find the audiobooks via your local library or have an Audible account, they are VERY well performed! Semi-spoiler: a female character becomes quite important in the second book & about half the novel is told from her viewpoint. Maybe try the audiobook for the first novel to entice them into reading the other two in the series? 
The Hunger Games trilogy is also quite good in either audiobook or regular-book form; and with a movie based on the first novel coming out in March, that may be impetus for them to read the series.
***The Forest of Hands and Teeth *** was also fairly good - a bit angsty for my tastes now, but probably would have appealed more to me as a teen. Haven’t read the follow-up books (yet)
I’m almost ashamed to admit this, but I kinda enjoyed I Kissed a Zombie, and I Liked It by Adam Selzer - the title intrigued me enough to get it from the library. It’s fluffy and definitely aimed toward teen girls, but had some fun turns of phrase and an interesting take on the zombie genre.
If you/they are open to urban fantasy, I found Hunger - by Jackie Morse Kessler interesting - a young woman dealing with anorexia is assigned as the incarnation of Famine, one of the 4 Horsemen of the Apocalypse. I haven’t read the second book Rage, yet - and the other 2 are still in the pipeline.
A friend of mine runs a book review blog: http://doubleshotreviews.com/ that covers a lot of genre novels, including zombie/post apocalyptic stuff.
I’m in the middle of Divergent and enjoying it. It takes place in Chicago. Teenagers are required to choose a faction in which to live. It’s fairly new.
Agreed that it’s better than The Stand, but bear in mind, it’s a bit dated, very 1985 Cold War-esque.
Joe
I was going to suggest this. I read it as a little kid and loved it but then I read it again as a teenager and thought it was kinda crap. A young teen might enjoy it.
I have picked up several of these for Xmas now, including The Forest of Hands and Teeth. I’m looking forward to reading some of them myself.
I read that recently, too, and I enjoyed it quite a bit.
Chiming in to say that I enjoyed Divergent as well (sequel in May: Insurgent , and I also quite liked the interesting take on Hunger and Rage by Jackie Kessler. Pretty interesting stuff, but some heavy moralizing going on. (not exactly ***controversial ***moralizing, but still.)
Another few I thought of that are more dystopian than zombies:
When She Woke is sort of a fan-fic feeling mashup of The Handmaid’s Tale and The Scarlet Letter. If teen person likes it, she may try out either of the following and be all meta and aware of the subtext and stuff. 
Nomansland is a fun little read - really short. Post apocalypse, all the women get an island, and the ranger protagonist joins up with a clique of girls investigating remains from Before - a major crime. Lots of good stuff in there, with crazy people and adults with hidden motives and lots of secrets that everyone keeps from everyone else.
Getting into the recent spate of OMG feminist teens books, we have the post-apocalyptic, post-diaspora, generation-spaceship-riding, or otherwise non-standard future world inhabiting girl protagonists who fight sexism, forced marriages or birthings, or just general totalitarianism: (in no particular order)
Incarceron and sequel Sapphique
the very similar Wither and sequel (out in Feb) Fever
Ashes (back around to zombies!)
Glow which I was excited about because it’s a generation-ship novel, and I think those are quite interesting.
the list could go on, but those are just the ones I’ve read myself recently and enjoyed enough to suggest. 
Seconding Hunger Games in case it gets lost in the shuffle - popular yes; quite good, also and fits your requirements. First in a trilogy. I couldn’t put them down.
I absolutely concur with the Feed, Shipbreaker and When she woke recommendations from upthread. I just started reading Legend - I’m not that far into it, but have heard absolutely fantastic things about it and I’m liking it so far. Also, not zombies, but Post-A - Birthmarkedand Prizedby Caragh O’Brien
I also recommend Forever Young Adult for book reviews and suggestions - take a look at their Zombieand Dystopiacategories for possible suggestions.
One of my all-time favorite books.
Ah, dammit, I’m trying to recommend a generation ship novel for you that I recently read, and I can’t for the life of me remember the title or author. Can anyone help me out? It starts out with the story of how one guy devotes his life to building this massive ship, and then goes on to tell the stories of generations of people living in the ship. The ship is colonized by regular people and a religious faction, which turns into a big war on the ship eventually. It ends with the main character designing wings so he fly up into the areas that used to be populated by the scientists who built the ship, but have been lost in antiquity.