In his message, Mesic expressed a wish for the further promotion of good and friendly relations between the two countries.
Peres became the ninth Israeli president after receiving 86 out of 120 votes in the Knesset. He was elected in the second round after the other two candidates withdrew from the race and gave him their support.
Peres will be sworn in next month when the seven-year term of the incumbent President, Moshe Katsav, formally ends.