The men, awaiting trial for burglary, were found missing when they failed to show up for roll call on Sunday.
Officials said one of the fugitives had been captured and was being questioned.
The jail, thought to be Asia's largest, houses some 13,500 prisoners, including many of India's most feared militants, gangsters and other criminals.
This is the first time inmates have dug their way out of Tihar, reports said.
In what appears to be a carefully planned prison break reminiscent of the Hollywood film Shawshank Redemption, the two men - Faizan and Javed - are believed to have scaled the wall of one jail building to reach another where they dug a tunnel under the perimeter wall and crawled out through a drain, the Times of India reported.
Officials say they are still trying to piece together exactly how the escape was executed.
"They escaped sometime on Saturday or Sunday. We will find out how they  managed to escape. It is too early to say how and when they staged the jail break," news agency AFP quoted Tihar's deputy inspector general Mukesh Prasad as saying.
The prison has a maximum capacity of 6,250 prisoners but is massively overcrowded.
Officials said an investigation had been ordered to find out how the prisoners managed to slip past armed guards and electronic surveillance.
Two prisoners have escaped from India's maximum security Tihar jail in Delhi by digging a tunnel under a wall and scaling another, authorities said.