java.time
The java.util Date-Time API and their formatting API, SimpleDateFormat are outdated and error-prone. It is recommended to stop using them completely and switch to the modern Date-Time API*.
Also, quoted below is a notice from the home page of Joda-Time:
Note that from Java SE 8 onwards, users are asked to migrate to java.time (JSR-310) - a core part of the JDK which replaces this project.
Solution using java.time, the modern Date-Time API:
import java.time.Instant;
import java.time.ZoneId;
import java.time.ZonedDateTime;
public class Main {
    public static void main(String[] args) {
        long millis = 1316391494L;
        Instant instant = Instant.ofEpochMilli(millis);
        System.out.println(instant);
        // The same instant at a specific timezone
        ZonedDateTime zdt = instant.atZone(ZoneId.of("Australia/Brisbane"));
        System.out.println(zdt);
    }
}
Output:
1970-01-16T05:39:51.494Z
1970-01-16T15:39:51.494+10:00[Australia/Brisbane]
ONLINE DEMO
Learn more about the modern Date-Time API from Trail: Date Time.
What went wrong with your code?
A java.util.Date object simply represents an instant on the timeline — a wrapper around the number of milliseconds since the UNIX epoch (January 1, 1970, 00:00:00 GMT). Since it does not hold any timezone information, its toString function applies the JVM's timezone to return a String in the format, EEE MMM dd HH:mm:ss zzz yyyy, derived from this milliseconds value. To get the String representation of the java.util.Date object in a different format and timezone, you need to use SimpleDateFormat with the desired format and the applicable timezone e.g.
import java.text.SimpleDateFormat;
import java.util.Date;
import java.util.Locale;
import java.util.TimeZone;
public class Main {
    public static void main(String[] args) {
        long millis = 1316391494L;
        Date date = new Date(millis);
        SimpleDateFormat sdf = new SimpleDateFormat("yyyy-MM-dd'T'HH:mm:ss.SSSXXX[zzzz]", Locale.ENGLISH);
        sdf.setTimeZone(TimeZone.getTimeZone("Etc/UTC"));
        String strDateUtc = sdf.format(date);
        System.out.println(strDateUtc);
        sdf.setTimeZone(TimeZone.getTimeZone("Australia/Brisbane"));
        String strDateBrisbane = sdf.format(date);
        System.out.println(strDateBrisbane);
    }
}
Output:
1970-01-16T05:39:51.494Z[Coordinated Universal Time]
1970-01-16T15:39:51.494+10:00[Australian Eastern Standard Time]
ONLINE DEMO
* For any reason, if you have to stick to Java 6 or Java 7, you can use ThreeTen-Backport which backports most of the java.time functionality to Java 6 & 7. If you are working for an Android project and your Android API level is still not compliant with Java-8, check Java 8+ APIs available through desugaring and How to use ThreeTenABP in Android Project.