Appleton took charge of Pompey yesterday after leaving his job as Hodgson's No2.
Hodgson said: "I, and the rest of the coaching staff, would like to wish Michael all the very best in his new job and thank him very much for the work he has put in during my brief time at the club.
"I realise what an excellent club servant he has been over the years and he fully deserves this opportunity to manage a team of his own.
"We will miss him and, in an ideal world, we would have liked him to still be here with us.
"However, I was fully behind the club's decision not to stand in his way when this excellent opportunity presented itself and our loss is going to be Portsmouth's gain."