Pujara made 203 -- his third century in three matches for Sussex -- with wicketkeeper Rizwan marking his home debut for the south coast club with 79.
The pair put on 154 for the sixth wicket at Hove as Sussex piled up an imposing 538 in reply to Durham's first innings 223 in the Second Division of English cricket's red-ball County Championship.
Durham, however, ended the third day of four with hopes of holding out for a draw at 169 for none in their second innings.
That Pujara and Rizwan are both at Sussex is a rare example of Indian and Pakistani cricketers turning out for the same team.
India cut bilateral cricket ties with Pakistan after attacks in Mumbai in 2008 that authorities blamed on Pakistan and the teams now only play each other in multi-nation events.
The South Asian rivals have not faced each other in a bilateral series since Pakistan toured India in 2012-13 for a set of one-day internationals and T20 matches during a brief thaw in relations.
Pakistan's stars also do not play in the Indian Premier League (IPL), the world's most high-profile T20 competition, which is currently taking place in India.
