What is the right way to extend a furnace intake pipe under a brick patio?
I do not think extending it under the patio is "the right way" to do it. 

If rerouting it from the source up through the house is not an option then -

I would put a 90∘ elbow on it and run it up the  exterior wall a bit. Build a chase around it (box it in to keep it from getting crushed or damaged), similar to the one under patio. 

The elbows have male and female ends, the male end should go into the existing pipe, but you will not be able to get screws in it so adhesive will have to do. Add another elbow and a screen. 

Put a tilted cap on the new chase long enough for rainwater to drain off without getting sucked into the intake (similar to the the one in your photo of the final vent.) You could build a little removable wall shelve/table over it for your patio living space. 

Putting it under the patio seems like an expensive way to ensure more expense and problems down the road.