The prophet Isaiah penned the familiar words "Surely He hath borne our griefs and carried our sorrows."  Many composers have put these words to music, probably most famously G.F. Handel as part of the oratorio Messiah.
I've been thinking recently about whether the converse is true as well. If He carried our pain and sorrows, does He carry our joy as well?And he shall go forth, suffering pains and afflictions and temptations of every kind; and this that the word might be fulfilled which saith he will take upon him the pains and the sicknesses of his people. And he will take upon him death, that he may loose the bands of death which bind his people; and he will take upon him their infirmities, that his bowels may be filled with mercy, according to the flesh, that he may know according to the flesh how to succor his people according to their infirmities. 
I think He does. I went searching in the scriptures for support for this idea, and I didn't find anything directly on point. I found several passages that refer to God's joy over repentant sinners, but nothing about joy over our joys.
But I still think it's true. Jesus loves us and wants us to be happy. When people I love are happy, I feel happy, too. If I, as an imperfect person, feel that way, then how much more would our perfect and loving Savior feel?
The scriptures teach that joy is the purpose of our existence.  So I think when we achieve that, we make God happy.
 Isaiah 53:3
 Alma 7:11-12
 See e.g. 2 Nephi 2:25