Sure. Essentially, this is just a s/// operator. I've used angle brackets instead of slashes as the delimiter.

$str =~ s< 
           (           # Start capturing (to $1)
             \s*       # Optional whitespace (forgot this before)
             \"        # starting with a quote
             [^\"]+    # 1 more of anything except a quote
             \"        # another quote
             \s*       # optional white space
           )           # end capture
           (,|$)       # capture comma seperator (or EOL*)
         >
         <  # /e option makes this becomes a code block
            # put the contents of $1 into a localised writable var
            #($1 is read only!)
            # and use the tr/// op to  remove the commas
            (local $a = $1) =~ tr/,//d;  
            # The last expression in the block is used as the 
            # replacement expression, so we put the decomma'd localise
+d var here.
            # not forgetting to put back the final comma
            # If it was there*
             $a .$2 ; 
         >gex; #!" Global, Execute, ignore whitespace in regex
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The weird comment (#!") I forgot to remove, is just used to keep the syntax parser in my editor happy. Sorry for any confusion it caused.

Update* Added to correct error spotted by LAI. Use Text::CSV instead.