Radwanska beat Kaia Kanepi of Estonia 6-2, 6-4, and Cornet rallied to defeat Maria Kirilenko of Russia 1-6, 6-2, 6-1.
The 10th-ranked Radwanska has won three WTA Tour titles this season and is competing to qualify for the season-ending Sony Ericsson Championships.
The Pole broke Kanepi in the opening game of both sets and stayed in control throughout.
Radwanska, who had a bye in the first round, will next play either fifth-seeded former champion Nadia Petrova or Katarina Srebotnik of Slovenia.
Cornet struggled with her serve in the opening set and was broken twice by Kirilenko, before holding serve throughout the last two sets.
Flavia Pennetta continued her recent run of good results by beating Agnes Szavay of Hungary 6-1, 6-2 in a first-round match.
The seventh-seeded Italian, who reached the final of the Zurich Open last week, dropped only three points on serve in the opening set and broke Szavay on six occasions.
Sybille Bammer of Austria also advanced after beating Francesca Schiavone of Italy 7-5, 6-3 to set up a second-round match against top-seeded Ana Ivanovic.
Bammer was 5-4 down in the first set before winning five straight games. She added another break to close out the match.
Petra Kvitova of Czech Republic defeated Tamira Paszek 6-3, 6-3 to reach the second round.
Madrid Internationals Back In Training
Radwanska reaches quarterfinals in Linz
