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After Martinez spoke to the media first following his side's 3-1 victory, Mourinho interrupted the Spaniard as he addressed another group of journalists.
The Portuguese felt his interviews should have taken place first.
"Roberto, next time tell me to go before you because we have to travel," said Mourinho, who swore as he left.
Martinez replied: "I don't control that, Jose. I don't control that."
After Mourinho left abruptly, Martinez said to the journalists, including a Liverpool Echo reporter: "When he beat us 6-3 last season he was such a nice man. I prefer him like that."
He also walked out of a pre-match interview with BT Sport's Ray Stubbs when asked about the club's failed summer-long pursuit of Toffees defender John Stones, 21.
"Before the game this is too long," said Mourinho.
Defending champions Chelsea are 17th in the table, 11 points behind leaders Manchester City, having lost three of their past four games.
Mourinho said after Saturday's game: "I am the man for the job. I don't think there is a better man who could come and do my job."
Earlier this season, he was criticised for his treatment of the Blues' medical staff, having downgraded team doctor Eva Carneiro's role.
Mourinho said Carneiro and head physio Jon Fearn were "impulsive and naive" and did not "understand the game" after they went on the pitch to treat Eden Hazard during the 2-2 home draw with Swansea on 8 August.
Chelsea manager Jose Mourinho argued with Everton boss Roberto Martinez over managerial etiquette after his side lost at Goodison Park on Saturday.