According to UDI one of the ways you qualify as a skilled worker : special qualifications that you have obtained through long work experience, if relevant in combination with courses etc. A permit is only granted in such cases in exceptional circumstances. Your qualifications must be equivalent to those of someone who has completed vocational training [1].
You also need to show that you can support yourself for 6 months (NOK 119 392)
[0] https://www.udi.no/en/want-to-apply/work-immigration/job-see...
I was in a pretty much similar situation to you when I started to think about moving to Europe (or Canada, btw): some work experience; no formal degree; late twenties. I also felt there is some chicken and egg problem, though the solution is simple: just send your CV to interesting companies, asking if they are offering relocation from outside EU. Many large companies like Spotify, Klarna offer relocation, as well as meny smaller companies.
As you have went through recruitment process (which in Sweden is quite humane) and accepted the offer, your employer would have applied on behalf of you for work visa (typically 2-years, which later could be extended unless you get a permanent right of residence after four years).
If a company is willing to sponsor your visa then it's really not hard over here.
After four years in Sweden you would be granted a permanent residence, and one year later you could apply for Swedish passport.
The requirement of 'degree in computer science' is just a part of standard job description, at least I was never asked about my (missing) degree.
One interesting option to consider are 'consultant agencies'. They hire you on a permanent basis (often offering relocation) and then offer you to companies. This way you may reach companies not offering relocation themselves (e.g. startups). A small consultancy I am working for now, Go See Talents [2], is offering relocation from outside EU (along with mentoring for less experienced developers, which is probably not the case for you)
[1] https://www.arbetsformedlingen.se
[2] http://goseetalents.com