Is there really no concept of low-cost airline over there? Or is it just an omission?
The list you mention displays low-cost airlines by their home country. This is in principle the country where the airline's headquarters are. For instance, Ryanair is listed under Ireland. This does not mean that the company only flies to and from Ireland. It has bases and destinations all over Europe and in Northern Africa. No entry for Azerbaijan merely means that the authors of this Wikipedia entry have not identified any low-cost company based in Azerbaijan. It does not mean that there is no low-cost company flying to Azerbaijan
Moreover, the concept of "low-cost carrier" is not precisely and unambiguously defined. This means that if you are asking several persons what a low-cost airline is, you will get several opinions. Anyway, it is only terminology. What matters is the price you have to pay for a certain service.
To get a more definitive answer, have a look at the homepage of Baku airport. There you will find a list of companies operating from there.
The only international low-cost airline flying to Schengen Area from Azerbaijan is Wizz Air:
Baku - Budapest (each Monday and Thursday) and Budapest - Baku (each Wednesday and Sunday) flights are available to book for as much as 50 - 80 EUR (one-way). From/to Budapest you can travel to/from dozens of destinations in Europe with Wizz Air. As compared to local non-low-cost airline AZAL, the prices offered by Wizz Air are generally twice or even as much as three times cheaper.
The newest local low-cost airline which is part of AZAL, Buta Airways started operating from September 2017. Those are must-chose tickets (starting from 29 EUR*, one-way) when travelling by air from/to Azerbaijan to/from Sofia (Bulgaria), Istanbul (Turkey), Tbilisi (Georgia), Moscow, St. Petersburg (Russia), Kyiv (Ukraine) and Tehran (Iran).
*Note: Neither checked nor cabin baggage is included in this price
Also a low-cost airline which is part of AZAL. AZALJET started operating from March 2016.
Flights to the most popular destinations from/to Baku with duration of up to 3.5 hours (excluding the ones operated by Buta Airways) are being operated under the AZALJET brand. These destinations include Dubai, Minsk, Lviv, Aktau, Ankara and the Turkish Izmir, Bodrum and Dalaman resorts. Ticket prices start from 49 EUR. AZALJET doesn't have its own website rather than the tickets are directly sold at AZAL website.
There were also cheap Pegasus flights from Istanbul, Turkey to Gabala, Azerbaijan (and vice-versa), but they left Azerbaijani market in March 2016. (Most probably because the same route has been opened by AZALJET and lately has been taken by Buta Airways)
AzAL would probably not classify itself as a low cost airline, but offers very good rates to destinations in the region. http://www.azal.az/
There are several low cost airlines that serve the region. But keep in mind that the South Caucasus is quite a distance away from Europe, if that's where you're coming from. Meaning that the cost of flying there will be a bit more than for other intra-Europe flights.
Check http://airninja.com for low cost airlines that serve your destinations.
By the latest update, Azal began providing AzalJet as lowcost provider. Even in Azal's website you can see the flight. There are flights both from Baku (capital) and Gabala (city in the middle of Azerbaijan).