There's advice from Joel Spolsky, beloved co-founder and ex-CEO of the StackExchange network. He has written blog articles with names like:
You can read those and present some of the arguments to your boss. Mr Spolsky's company sold its software Trello to Atlassian for >400.000.000$, so he seems to know a thing or two about managing a software company.
No matter how hard it is to find testers, they are still cheaper than programmers. A lot cheaper. And if you don’t hire testers, you’re going to have programmers doing testing. And if you think it’s bad when you have testers churning out, just wait till you see how expensive it is to replace that star programmer, at $100,000 a year, who got sick of being told to “spend a few weeks on testing before we release” and moved on to a more professional company. - Joel Spolsky, reason 5.
[Emphasis mine]
However, even if you do manage to change your company's way of working, this won't happen overnight. You'll have to do it this time. Remember that you get money in exchange for it, if that helps.