The Rashtriya Janata Dal's Suday Yadav defeated Janata Dal-United candidate Abhiram Sharma by 35,036 votes in Jehanabad.
A formal announcement of the two victories will be made soon, the officials said.
In Araria, RJD candidate Sarfaraz Alam was leading by more than 60,000 votes over the BJP's Pradeep Kumar Singh.
RJD leaders and workers celebrated across the state by distributing sweets and bursting fire crackers.
The by-elections were held on Sunday.
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