No, Arjuna did not kill Dronacharya.. You might be thinking of Bhishma. In any case, here's how this chapter of the Drona Parva of the Mahabharata describes Yudhishthira's lie:
Hearing those words of Bhima and induced by the counsels of Krishna, and owing also to the inevitability of destiny, O monarch, Yudhishthira made up his mind to say what he desired. Fearing to utter an untruth, but earnestly desirous of victory, Yudhishthira distinctly said that Aswatthaman was dead, adding indistinctly the world elephant (after the name), Before this, Yudhishthira's car had stayed at a height of four fingers' breadth from the surface of the earth; after, however, he had said that untruth, his (vehicle and) animals touched the earth. Hearing those words from Yudhishthira, the mighty car-warrior Drona, afflicted with grief, for the (supposed) death of his son, yielded to the influence of despair. By the words, again, of the Rishis, he regarded himself a great offender against the high-souled Pandavas. Hearing now about the death of his son, he became perfectly cheerless and filled with anxiety; upon beholding Dhrishtadyumna, O king, that chastiser of foes could not fight as before.
And here is how this later chapter of the Drona Parva describes Dronacharya's death:
Then Bhima, of great wrath, holding the car of Drona, O monarch, slowly said these words unto him: If wretches amongst Brahmanas, discontented with the avocations of their own order, but well-versed in arms, did not fight, the Kshatriya order then would not have been thus exterminated. Abstention from injury to all creatures hath been said to be the highest of all virtues. The Brahmana is the root of that virtue. As regards thyself, again, thou art the foremost of all persons acquainted with Brahma. Slaying all those Mlecchas and other warriors, who, however, are all engaged in the proper avocations of their order, moved thereto by ignorance and folly, O Brahmana, and by the desire of wealth for benefiting sons and wives; indeed, for the sake of an only son, why dost thou not feel ashamed? He for whom thou hast taken up weapons, and for whom thou livest, he, deprived of life, lieth today on the field of battle, unknown to thee and behind thy back. King Yudhishthira the just hath told thee this. It behoveth thee not to doubt this fact.' Thus addressed by Bhima, Drona laid aside his bow. Desirous of laying aside all his weapons also, Bharadwaja's son of virtuous soul said aloud, 'O Karna, Karna, O great bowman, O Kripa, O Duryodhana, I tell you repeatedly, exert yourselves carefully in battle. Let no injury happen to you from the Pandayas. As regards myself, I lay aside my weapons.' Saying these words, he began loudly to take the name of Aswatthaman. Laying aside his weapons then in that battle, and sitting down on the terrace of his car, he devoted himself to Yoga and assured all creatures, dispelling their fears. Beholding that opportunity, Dhrishtadyumna mustered all his energy. Laying down on the car his formidable bow, with arrow fixed on the bow-string, he took up a sword, and jumping down from his vehicle, rushed quickly against Drona. All creatures, human beings and others, uttered exclamation of woe, beholding Drona thus brought under Dhrishtadyumna's power. Loud cries of Oh and Alas were uttered, as also those of Oh and Fie. As regards Drona himself, abandoning his weapons, he was then in a supremely tranquil state. Having said those words he had devoted himself to Yoga. Endued with great effulgence and possessed of high ascetic merit, he had fixed his heart on that Supreme and Ancient Being, viz., Vishnu. Bending his face slightly down, and heaving his breast forward, and closing his eyes, and resting ort the quality of goodness, and disposing his heart to contemplation, and thinking on the monosyllable Om, representing. Brahma, and remembering the puissant, supreme, and indestructible God of gods, the radiant Drona or high ascetic merit, the preceptor (of the Kurus and the Pandavas) repaired to heaven that is so difficult of being attained even by the pious. Indeed, when Drona thus proceeded to heaven it seemed to us that there were then two suns in the firmament. The whole welkin was ablaze and seemed to be one vast expanse of equal light when the sun-like Bharadwaja, of solar effulgence, disappeared. Confused sounds of joy were heard, uttered by the delighted celestials. When Drona thus repaired to the region of Brahman, Dhrishtadyumna stood, unconscious of it all, beside him. Only we five amongst men beheld the high-souled Drona rapt in Yoga proceed to the highest region of blessedness. These five were myself, Dhananjaya, the son of Pritha, and Drona's son, Aswatthaman, and Vasudeva of Vrishni's race, and king Yudhishthira the just, the son of Pandu. Nobody else, O king, could see that glory of the wise Drona, devoted to Yoga, while passing out of the world. In fact, all human beings were unconscious of the fact that the preceptor attained to the supreme region of Brahman, a region mysterious to the very gods, and one that is the highest of all. Indeed, none of them could see the preceptor, that chastiser of foes, proceed to the region of Brahman, devoted to Yoga in the company of the foremost of Rishis, his body mangled with arrows and bathed in blood, after he had laid aside his weapons. As regards Prishata's son, though everybody cried fie on him, yet casting his eyes on the lifeless Drona's head, he began to drag it. With his sword, then, he lopped off from his foe's trunk that head,--his foe remained speechless the while. Having slain Bharadwaja's son. Dhrishtadyumna was filled with great joy, and uttered leonine shouts, whirling his sword.
So as you can see, Dronacharya already left his body before Drupada's son Drishtadyumna cut off his head.
