The move came after Satpathy's recent remarks on Mishra, which created ripples in the state.
Both Satpathy and Mishra met Patnaik, the BJD president, in an apparent bid to bury the issue and show unity in the party.
Also present at Naveen Nivas, the chief minister's residence, were BJD spokesperson and new Rajya Sabha member Pratap Keshari Deb and former minister Pranab Prakash Das.
Stating that he has much respect and affection for Debi Mishra, Satpathy said what he had stated (about Mishra) was a slip of tongue.
"We are good friends and there are no differences between us," the Dhenkanal MP said after meeting Patnaik.
The senior BJD leader said he has already expressed regret over his remarks.
Speaking in a similar tone, Debi Mishra said "We are close friends. He (Satpathy) has already clarified in public domain."
Satpathy, while addressing a public meeting in his home constituency on June 7, had stated that Patnaik had removed some leaders from the ministry due to their wrong doings. He had also named Debi Mishra, who was appointed vice-president of BJD, after he resigned from the ministry.
Satpathy subsequently expressed regret over his remarks. "Regret shooting my mouth and hurting sentiments. Will try to improve my characteristic flaw," Satpathy had tweeted.
The chief minister had also maintained that his act of giving some senior leaders important party positions after removing them as ministers last month should not be construed otherwise.
"I have given a great deal of important political work to certain leaders of our party. Nobody should construe it as otherwise," Patnaik had said.