The machine, launched in March, can be used as both a home console and a portable device, with detachable controllers that can be used in several configurations.
The share price of the 127-year-old business hit ¥31,880 (£220), surpassing the level it briefly reached last summer as augmented reality game Pokémon Go hit the peak of its popularity.
It is now at its highest level since April 2010 – around the point that sales of the Wii began to tail off.
The Wii had opened up home gaming to new audiences, including families and older adults, with its innovative motion controls. It sold more than 100 million consoles and became the only Nintendo home console in the last 20 years to outsell its competitor from Sony, the PlayStation 3.
But Nintendo failed to replicate this success with its next device, the Wii U, launched in 2012.
The Switch, however, sold 2.74 million units in its first month, and has sold out nearly everywhere since launch, The Guardian reports. Nintendo president Tatsumi Kimishima said that if it hit 10 million sales by the end of the year, the Switch would be on track to become bigger than the Wii.