It is common to simply swap alphabet for use in urls, so that no %-encoding is necessary; only 3 of the 65 characters are problematic - +, / and =. the most common replacements are - in place of + and _ in place of /. As for the padding: just remove it (the =); you can infer the amount of padding needed. At the other end: just reverse the process:
string returnValue = System.Convert.ToBase64String(toEncodeAsBytes)
        .TrimEnd(padding).Replace('+', '-').Replace('/', '_');
with:
static readonly char[] padding = { '=' };
and to reverse:
string incoming = returnValue
    .Replace('_', '/').Replace('-', '+');
switch(returnValue.Length % 4) {
    case 2: incoming += "=="; break;
    case 3: incoming += "="; break;
}
byte[] bytes = Convert.FromBase64String(incoming);
string originalText = Encoding.ASCII.GetString(bytes);
The interesting question, however, is: is this the same approach that the "common codec library" uses? It would certainly be a reasonable first thing to test - this is a pretty common approach.