This incident is mentioned in Udyoga Parva of Mahabharata. First some clarification to your question.
If I remember correct Duryodhan arrives first but since the Lord was asleep he sits near his feet. Arjun however arrives after but stands next to the lord.
Yes, Duryodhana arrived first and sat near the head of Lord Krishna and Arjuna arrived second and stood at the back of bed.
And as Krishna was sleeping, Duryodhana entered the room, and sat down on a fine seat at the head of the bed. And after him entered that wearer of the diadem the magnanimous Arjuna. And stood at the back of the bed, bowing and joining his hands. And when the descendant of Vrishni, Krishna awoke, he first cast his eyes on Arjuna. And having asked them as to the safety of their journey, and having fitly bestowed his greetings upon them, the slayer of Madhu questioned them as to the occasion of their visit.
But Duryodhan arrived first so what was the reason behind the lord being biased towards Arjun?
Lord Krishna was not biased in this situation. Since, he has seen Arjuna first and also Arjuna is junior to Duryodhana in age so, Arjuna got priority.
Then Duryodhana addressed Krishna, with a cheerful countenance, saying, It behoveth you to lend me your help in the impending war. Arjuna and myself are both equally your friends. And, O descendant of Madhu, you also bear the same relationship to both of us. And today, O slayer of Madhu, I have been the first to come to you. Right-minded persons take up the cause of him who comes first to them. This is how the ancients acted. And, O Krishna, you stand at the very top of all right-minded persons in the world, and are always respected. I ask you to follow the rule of conduct observed by rightminded men.
' Thereat Krishna replied, 'That you have come first, O king, I do not in the least doubt. But, O king, the son of Kunti, Dhananjaya, has been first beheld by me. On account of your first arrival, and on account of my having beheld Arjuna first, I shall, no doubt, lend my assistance, O Suyodhana, to both. But it is said that those who are junior in years should have the first choice. Therefore, Dhananjaya, the son of Kunti, is entitled to first choice. There is a large body of cowherds numbering ten crores, rivalling me in strength and known as the Narayanas, all of whom are able to fight in the thick of battle. These soldiers, irresistible in battle, shall be sent to one of you and I alone, resolved not to fight on the field, and laying down my arms, will go to the other. You may, O son of Kunti, first select whichever of these two commends itself to you. For, according to law, you have the right to the first choice.'
'Thus addressed by Krishna, Dhananjaya the son of Kunti selected Kesava who was not to fight on the battle-field, even Narayana himself, the slayer of foes, increate, born among men at his own will,--the foremost of all Kshatriyas and above all the gods and the Danavas. And Duryodhana selected for himself that entire army (composed of the Narayanas). And, O descendant of Bharata, having obtained those troops numbering thousands upon thousands, he was exceedingly delighted, although he knew that Krishna was not on his side.
And at last Duryodhana was delighted and very happy that he has got the army of Krishna.