The 28-year-old Slovakian successfully underwent a medical and his new employers announced his arrival earlier today.
However, Boro would not confirm the former Inter Bratislava player's departure until the documentation was completed this evening.
'We wish 'Szil' the best of luck in his future career,' said manager Steve McClaren.
'He was a valuable squad member for me throughout the past four years or so, scoring some vital goals along the way.
'However, I wasn't able to guarantee him regular first-team football, so it was best for Szil to move on to pastures new.'
Nemeth, who made 146 appearances for the Teessiders, was signed by previous manager Bryan Robson in tandem with Terry Venables, but did not arrive until after McClaren had taken over in the summer of 2001.
'In England, my situation had become difficult,' the player told Strasbourg's official website, www.rcstrasbourg.fr.
'Middlesbrough have a lot of players for each area.
'There are a lot of players and only 11 are lined up at the weekend. The manager preferred to give their chance to the newcomers. That's football's law.'
Nemeth, who has cost a 'nominal fee', could make his debut in Saturday's match against Paris St Germain.
'I have not played in a while, but I passed my medical and the results were good,' he said.
'I am ready to play against Paris on Saturday.
'I am aware of the team's difficulties, but I don't want to put myself under any extra pressure. I will do my best to rediscover my top level quickly.
'I know there is no time to waste. I just want to adapt fast and score fast too.'