Salmond condemns Scotland's religious divide
A fan attacked Celtic manager Neil Lennon at a match against Hearts in Edinburgh last week and Lennon was also sent a parcel bomb last month.
"Modern Scotland is built on equality," Salmond told the Scottish parliament on Wednesday.
"We will not tolerate sectarianism as a parasite in our national game of football or anywhere else in this society," he added after being appointed first minister in Scotland's devolved parliament.
Celtic and Glasgow rivals Rangers dominate Scottish domestic football, but their on-field rivalry is marred by religious bigotry and sectarian violence.
Rangers have a predominantly Protestant fan base, while Celtic draw much of their support from the Catholic community in western and central Scotland. The sectarian divide mirrors and feeds off the divisions that have plagued Northern Ireland.
Lennon, a former Northern Ireland international, called this week for an end to sectarian chanting after reports that some fans sang pro-Irish Republican Army (IRA) songs at last week's troubled game with Hearts.