To give an anecdotal response; the 840 Pro did work on the H310 when I tried them.
The problem is, especially the firmware of the H310 has a ridiculous low queue depth preset and because of this may (depending on your work load) suffer from a terrible I/O performance.
You have a chance to change that, through cross-flashing the controller to get rid of the PERC firmware. The controller is an OEM version of the LSI SAS 2008 and can be reflashed to LSI mode (queue depth 600 if I remember correctly). Another possibility is to get a proper controller (PERC H700/H710 for example) - those also work with the 840.
Here is a good comparison between various controller queue depths. The H310 has a queue depth of 25, while for example the H700 has a queue depth of 975.