The striker made it 23 goals in 56 caps to move ahead of previous record holder Niall Quinn - all at the age of 24.
It was always a case of when Robbie would break the record and it took just 14 minutes at Lansdowne Road as he buried a penalty following a foul on Damien Duff.
Robbie had Roy Keane to thank for his second goal after 33 minutes as the Manchester United midfielder squared the ball rather than go himself and Robbie tucked it away.
He could have had a hat-trick but was denied by Faroe keeper Jakub Mikkelsen on numerous occasions and when Mikkelsen was beaten, Jon Jacobsen who raced back to clear off the line.
Robbie Keane broke the Republic of Ireland goalscoring record with a double against the Faroe Islands on Wednesday night.