UEFA has charged the Bulgarian Football Union over racist behaviour by fans during England's 6-0 Euro 2020 qualifying win.
The game was halted twice before half-time and a group of supporters who made "monkey chants" and Nazi salutes were ejected from the ground.
Tuesday's fallout from those shameful scenes included BFU president Borislav Mihaylov resigning after Prime Minister Boyko Borissov threatened to cut funding if he did not step down.
FIFA pledged to extend any sanctions imposed by UEFA worldwide and Infantino, who previously fronted European football's governing body, suggested now is the time for football to take a harder line against racism, including life bans for any perpetrators.
"So many times we say there is no place for racism in football, but nonetheless we still face challenges to tackle this problem in our sport, as we do in society," he said.
"We will need the support of public authorities to help us identify and punish the culprits but we probably also need to think more broadly on what we can do to fix this.
"When we proposed the three-step procedure in 2009 when I was at UEFA, and then made the regulations even tougher a few years later, we could not have imagined that so shortly thereafter we would again be having to think of how to combat this obnoxious disease that seems to be getting even worse in some parts of the world."
Infantino said: "I call on all football governing bodies to join us and think together of new, stronger and more effective ways to eradicate racism in football.
"As a starting point, I suggest that all competition organisers enact regulations which envisage life bans from stadiums for those who are found guilty of racist behaviour at a football match. FIFA can then enforce such bans at a worldwide level."
UEFA's three-step procedure to deal with racist incidents was partially enacted during Monday's match, with the initial stoppage coming after England players reported chants to the referee and an announcement calling for the abuse to cease was made over the stadium's public address system.
After a further complaint, match official Ivan Bebek asked England manager Gareth Southgate and captain Harry Kane whether he wished for them to take the teams from the field – in line with step two.
The close proximity to half-time was a factor in England being minded to play on and Southgate credited Bebek's conduct throughout as being "outstanding".
England debutant Tyrone Mings confirmed the players unanimously agreed to continue playing at half-time. The third step in the UEFA plan after taking the players from the field is an abandonment if abuse persists.
Infantino's UEFA successor Aleksander Ceferin made a strong defence of his organisation's record when it comes to dealing with racism.
"As a governing body, I know we are not going to win any popularity contests. But some of the views expressed about UEFA’s approach to fighting racism have been a long way off the mark," he said.
"UEFA, in close cooperation with the FARE network (Football Against Racism Europe), instituted the three-stage protocol for identifying and tackling racist behaviour during games.
"UEFA's sanctions are among the toughest in sport for clubs and associations whose supporters are racist at our matches."