The sanction however is only half the 10-match minimum ban being introduced by UEFA into European competitions from next season, and campaigners have labelled the FA's decision "a missed opportunity".
The FA's sanction will be the "entry level" punishment for the "least serious" discriminatory offences with longer bans for higher level offences. Players found guilty of offences will have to attend educational programmes, and for a second offence there will be a minimum 10-game ban.
Clubs who have two or more players found guilty of discrimination offences within a 12-month period will also face charges and could even have points deducted. Although there are no openly gay players in English professional football, the rules will apply to any player who uses homophobic abuse to any other player.
The new rules come after a review of sanctions following the high-profile cases which saw John Terry banned for four matches and Luis Suarez for eight games for racist abuse on the pitch. It is understood that under the new rules both of those cases would carry longer bans than just five games.
FA chairman David Bernstein said: "Importantly, today's agreement encompasses all elements of discrimination, not just racial abuse."
There is potential embarrassment for the FA however given that it is hosting UEFA's Congress in London next week, and Europe's governing body will submit a resolution recommending that all member associations follow its lead on 10 matches.
Piara Powar, executive director of European anti-racism body FARE and a member of FIFA's new anti-racism task force, said: "As someone working across borders to promote anti-discrimination I think it's a shame it can't be the 10 games UEFA say they will implement in their own competitions and that they are recommending all national associations will adopt."
Bernstein defended the FA's decision not to follow UEFA's lead, saying the five-game ban had been agreed by all parts of the English game including anti-racism body Kick It Out. He said: "From our point of view it [10-match ban] has no subtlety to it. It should have subtlety to it. Any racism is unacceptable but there are different levels of offence.
"It's also a timing issue. We have been through an extensive process and have to get it approved through English football. It [UEFA's ban] came in right at the end of the process when we have spent months getting a consensus. But if European football says the line is in the wrong place then we may have to re-evaluate that."