Sort by custom alphabet

Bash+coreutils, 37 bytes

tr ,$2 \\na-z<<<$1|sort|tr \\na-z ,$2

Output:

$ ./alphasort.sh home,oval,cat,egg,network,green bcdfghijklmnpqrstvwxyzaeiouy
cat,green,home,network,egg,oval, $ 

CJam, 26 19 17 bytes

rr:A;',/{Af#}$',*

Try it online.

Test case

$ cjam sort.cjam <<< 'home,oval,cat,egg,network,green bcdfghjklmnpqrstvwxzaeiouy'
cat,green,home,network,egg,oval

How it works

rr                    " Read two whitespace-separated tokens from STDIN. ";
  :A;                 " Save the second token (the alphabet) in A.       ";
     ',/              " Split the remaining token at commas.             ";
        {Af#}$        " Sort by the chunks' characters' indexes in A.    ";
               ',*    " Join, separating by commas.                      ";

Pyth, 19 characters

j\,o_mx_zdNchczd\,

Test:

$ pyth -c "j\,o_mx_zdNchczd\," <<< 'home,oval,cat,egg,network,green bcdfghjklmnpqrstvwxzaeiouy'
cat,green,home,network,egg,oval

Explanation:

                            Implicit: d=" "
                            Implicit: z=input()
j\,                         ",".join(
   o                                 order_by(lambda N:
    _                                                  rev(
     m                                                     map(lambda d:
      x_zd                                                              rev(z).index(d),
      N                                                                 N),
    chczd\,                                            z.split(" "[0].split(",")

Essentially, it sorts the chunks, with a key of the list of indexes of the characters in the string, then joins them on commas. The reversal businesses is shorter than spliting the string again.