it depends who you are talking to. If you are telling it to your friend"プレゼントありがとう！" is the best way to say it. if you wanna tell it to older people”プレゼントありがとうございます。" these are natural way to say it.

Your 贈り物をありがとうございます　is perfect. 贈り物 and プレゼント are interchangeable. So, like Ibuki-san wrote, プレゼントありがとう／プレゼントありがとうございます are ok too. プレゼントありがとう＜プレゼントをありがとう＜プレゼントありがとうございます＜プレゼントをありがとうございます＜プレゼントをくれてありがとうございます(more polite) You can omit を, but I think that might sounds bit curt.　　　 So if you'd like to say it politely, I recommend you to use を.