Happy, having fallen asleep start dreaming (7)
I believe the answer is:
pleased
'happy' is the definition.
(similar in meaning)
'fallen asleep start dreaming' is the wordplay.
'fallen' is an anagram indicator.
'start' indicates taking the first letters.
The initial letter of 'dreaming' is 'd'.
'asleep' anagrammed gives 'please'.
'please'+'d'='PLEASED'
'having' is the link.
(Other definitions for pleased that I've seen before include "like Punch?" , "Grateful, delighted" , "Gratified" , "Glad" , "Well satisfied" .)