The requirement, which was imposed by the Italian-speaking canton in 2015, will be lifted as part of a new tax agreement between the canton and Italy, Ticino's government said in a statement.
Quashing the controversial measure resolves the “last obstacle” to concluding the new tax agreement, it said.
The issue caused indignation in Italy when it was first imposed as part of a number of measures over the past few years aimed at reducing the number of Italian cross-border workers – known as frontalieri – in the Swiss canton.
Ticino has historically had high numbers of frontalieri. In 2016 they made up 27 percent of the canton's active workforce.
Many locals feel that has led to higher unemployment and lower salaries for Swiss workers due to so-called wage-dumping, when foreign workers are hired at a cut price.
In 2014 the canton raised taxes on frontalieri, and in September 2016 Swiss voters in the canton backed a popular initiative to give preference to local workers over foreigners, though it remains to be seen if the plan will be implemented.
Under the new agreement between Ticino and Italy, taxes on non-Swiss-resident Italian workers will be gradually increased (up to a limit of 70 percent), making it less attractive for them to work in the canton and helping to reduce wage-dumping.
Italian workers will be expected to show any criminal record on a voluntary basis, while the canton will reserve the right to ask for it in certain cases.
The cantonal government said it had fully assessed the pros and cons of the new agreement and decided that “Despite not fully meeting Ticino's expectations, the new tax agreement is globally positive for our canton”.
According to official figures, last year there were a total of 318,500 people working in Switzerland but living over the border, a 3.7 percent increase on the previous year.
Over half of those lived in France, a quarter in Italy and a fifth in Germany.
The Lake Geneva region had the highest number of cross-border workers in the country but they still only made up 12 percent of the area's job market, a much lower percentage than Ticino's 27 percent.