When the 39-year-old reached 98 in the third Test against New Zealand in Hamilton on Friday he moved to 11,175 runs to leap-frog the former Australian captain on the list.
He went on to make an unbeaten 122, his 29th century in his 153rd Test and now has 11,199 runs.
It was his 17th unbeaten century in Tests, moving ahead of Indian great Sachin Tendulkar.
Border retired in March 1994 after making 11,174 runs from 156 Tests.
Ahead of Chanderpaul on the list is fellow West Indian Brian Lara on 11,953, while the recently retired Tendulkar heads the pile with 15,921 runs.
Jacques Kallis, the only other active player in the top seven, is fourth on 13,140 after making a first ball duck in South Africa's current Test against India in Johannesburg.