The short answer is that they didn't; the Chinese had guns before them.
It isn't clear exactly when the Chinese first started using guns. In large part, this is the fault of the fire-lance. The fire-lance is a short gunpowder-filled tube on a stick, used as an incendiary weapon. Drawings of it in use show flames emerging from the end. Early guns were also short tubes on sticks, and drawings of them also show flames emerging from the end. To confuse matters more, there were weapons which combines incendiary and projectile effects, fire-lances with things like pieces of broken porcelain added that would be shot out along with the flame (Needham calls these "co-viative projectiles"). Of course, written descriptions of these weapons in use can be similar. Early Chinese guns and fire-lances are both called 銃 (chòng), or 火槍 (huǒqiāng, "fire spear"), so names of the weapons are not enough - sufficiently clear descriptions of them are needed. (There is a similar problem when trying to distinguish between Chinese catapults and cannons - both are called by the same name, and catapults were used to throw explosive gunpowder bombs (so explosions and gunpowder could involve either weapon).)
Drawing of fire-lance (with spear): https://commons.wikimedia.org/wiki/File:%E6%A2%A8%E8%8A%B1%E9%8E%97.jpg
Drawing of fire-lance with co-viative projectiles: https://commons.wikimedia.org/wiki/File:Coviatice_lance.jpg
Drawing of European fire-lance (being used as a lance!): https://commons.wikimedia.org/wiki/File:Fire_Lance_Knight.jpg
Drawing of gun (multi-barreled): https://commons.wikimedia.org/wiki/File:Organ_gun.jpg
Drawing of gun (European): https://commons.wikimedia.org/wiki/File:Stangenb%C3%BCchsen_bei_Burgebelagerung_1475.jpg
Multi-barreled gun: https://commons.wikimedia.org/wiki/File:10_shot_hand_cannon_%28handgonne%29.jpg
Gun (European): https://commons.wikimedia.org/wiki/File:Tannenbergb%C3%BCchse.jpg
If the weapon is described as being used for incendiary effect, it is a fire-lance. If it is being described as shooting projectiles through armour, it is a gun (the co-viatives in a fire-lance are low-velocity compared to bullets from a gun).
The fire-lance may have been in use in China as early as the mid-10th century. It was certainly in use in the 12th century. The gun was certainly in use in China in the mid-13th century (the Jin used guns against the Mongols in the siege of Kaifeng). Similar weapons appear in Europe about 200 years later than these certain dates.
There were improvements in guns that were made in Europe. The serpentine (an S or Z-shaped match-holder + trigger that allowed the gun to be held in both hands rather than one hand being needed to touch the match to the touch-hole) appears to have been invented in Europe c. 1400, and the true matchlock followed in the 2nd half of the 15th century. These improvements spread to both the Ottomans, and elsewhere in the Middle East, and China. China appears to have acquired the matchlock via the Ottomans. The serpentine was first used with early-type guns (short barrel on stick):
and led to the development of modern-looking guns like the arquebus and musket. Corned power was developed in Europe (coarse-grained gunpowder, which was easier to store, and more powerful), which contributed to the development of the arquebus and musket. (Corned powder might have been independently developed in China at about the same time.)
Bibliography:
Andrade, Tonio (2016), The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History, Princeton University Press.
Bert S. Hall, Weapons and Warfare in Renaissance Europe: Gunpowder, Technology, and Tactics, Johns Hopkins University Press, 2001.
Peter A. Lorge, The Asian Military Revolution: From Gunpowder to the Bomb, Cambridge University Press, 2008.
Needham, Joseph, Ho, P.-Y., Lu, G.-D., Wang, L. (1986), Science and Civilisation in China, Volume 5, Part 7: The Gunpowder Epic, Cambridge University Press.
