The PZPN's image was tarnished under Lato who received severe criticism for his uncompromising style and the lacklustre performances of the national team.
Sports Minister Joanna Mucha also notified prosecutors last November of a possible bribery case involving Lato. The former Poland striker denied the allegations and no charges were brought.
Boniek, who scored 24 goals in 80 appearances for Poland and had a brief stint as team coach in 2002, was one of Lato's chief critics.
"I realise what responsibility will now rest on my shoulders," the 56-year-old Boniek told reporters after gaining the votes of 61 of the 118 FA delegates in a ballot containing four other candidates.
"I can't compare this victory to what happened during my football career. Football and scoring goals gives happiness without thought.
"This victory forces me to do some serious thinking ... this is a different level of responsibility."
Lato and Boniek were in the Poland team that finished in third place at the 1982 World Cup in Spain.
Boniek left Widzew Lodz for Juventus in 1982 and helped the Italians win their first European Cup three years later.
After finishing his career at AS Roma he retired in 1988 and settled in Italy.