Fussy Baby

Hi all! I have a 13 days old baby boy and sonetimes he gets so fussy at night until dawn. He's not hungry or bloated naman. He just wants to be held and when we lay him in bed, he will start to cry again. My husband wants us to just let him cry it out and check on him without picking him up para hindi daw masanay. I tried to do this last night but my heart was really breaking. He continously cried for about 5 mins then evetually slept. I was so worried. Do you think we are doing the right thing?