His office said he wanted to apologise to the child and his family "for the hurt his words may have caused".
BBC reports the video also shows the Dalai Lama kissing the child on his lips.
"His Holiness often teases people he meets in an innocent and playful way, even in public and before cameras. He regrets the incident," his office said.
The footage has sparked widespread criticism, with many social media users saying it was inappropriate and disturbing.
The incident appears to have taken place at the Dalai Lama's temple in Dharamshala on 28 February. He had interacted with around 120 students who had completed a skills training programme organised by the M3M Foundation, the philanthropic arm of real estate company M3M Group.
The foundation uploaded photos from the event on social media in March - in one of them, the Dalai Lama is seen hugging the boy from the viral video.
In the video which has circulated online, the boy is seen asking if he can hug the Dalai Lama. The leader motions to his cheek, saying "first here" and the boy kisses his cheek and gives him a hug.
Then, while holding the boy's hand, the Dalai Lama motions to his lips and says "I think here also", and kisses the boy on the lips.
The leader then puts his forehead to that of the boy's, before sticking out his tongue, saying "and suck my tongue". As some people laugh, the boy sticks his tongue out before withdrawing a little, as does the Dalai Lama.
There are then more hugs, as the spiritual leader speaks to the boy for a while longer, telling him to look to "good human beings who create peace and happiness".
Sticking one's tongue out can be a form of greeting in Tibet.
The Dalai Lama has been living in exile in India since fleeing Tibet in 1959, following an uprising against Chinese rule there.
In 2019, the Dalai Lama's office apologised after the spiritual leader told the BBC in an interview that any future female Dalai Lama should be "attractive".