Yes. This is a common occurrence for visa'd workers in many countries. I've done it myself with a UK visa. Long story short, while your old passport may have expired, it doesn't mean the documents attached have. And if it's still the same, the UAE visa comes as separate A4 papers anyway?
Anyway, the general process would be to take both passports. You present the new one as your permit to travel, and the old one with the visa as your permit to enter the country to work.
Others online confirm this:
most of my colleagues carry around two passports - the current one and
the old one with the Dubai Residency Visa in them. I guess just
because the passport expires, doesn't mean the Dubai paperwork does.
And another one:
You will need to carry your old passport as well as your new one. Just
because a passport expires does not mean that any valid visas, work
permits and similar documents expire with it.