Varandas, who was in charge of the medical department from 2011 until earlier this year, finished ahead of former futsal goalkeeper Joao Benedito in the election after a passionate campaign which saw a record 22,510 votes cast by club members.
"It's the greatest mission of my life," said Varandas, the club's 43rd president, after the decision was announced in the early hours of Sunday.
Varandas will replace firebrand Bruno de Carvalho who was ousted in June in another vote by club members after a turbulent few months.
Following a Europa League defeat against Atletico Madrid in April, De Carvalho threatened to suspend almost the entire first-team squad and lashed out at the players, calling them “spoilt children”.
In May, the club's training ground outside Lisbon suffered an attack by hooded supporters, some armed with sticks, in which Dutch forward Bas Dost was injured and the dressing room was vandalised.
Five days later, Sporting were defeated by underdogs Desportivo Aves in the Portuguese Cup final.
Nine players quit in protest at the attack, citing just cause, and some of them accused De Carvalho of inciting anger among the fans. They included Portugal goalkeeper Rui Patricio who joined English Premier League club Wolverhampton Wanderers.
Others to leave included deep-lying midfielder William Carvalho and forwards Gelson Martins and Ruben Ribeiro, although Dost and midfielder Bruno Fernandes later decided to stay at the club.
Former AC Milan coach Sinisa Mihajlovic, hired just before De Carvalho was voted out, was sacked days later by the interim board. Jose Peseiro is now in charge of the team.
Sporting are one of Portugal’s big three clubs, along with Benfica and Porto, having won the Primeira Liga 18 times, although the last of those was in 2002.
Sporting have won three and drawn one of their first four league games of the season.
(Writing by Brian Homewood; editing by Clare Fallon)
