It's a combination of luck, inadequate accounting, and being in the right place at the right time.

Napoleon started his formal service in the French Army as a lieutenant of artillery but as soon as possible he got permission for a lengthy leave of absence that was common for most French officers. Here he would receive half pay until he his time was up. After this time period passed, he requested and was granted even more time off while he was settling family matters and helping fan the flames of a Corsican revolution that would eventually displace his family. By 1789, his second leave would pass and he would be written of as "finding another profession", suggesting that he wasn't going to return to service, but as you can see, 1789 was a very important and busy year.

After being driven out in 1791 by Paoli, a Corsicana hero of Napoleon's that was siding with the British, he returned to Paris to demand his rank and even suggested a promotion to colonel for serving in the Corsican part of the French National Guard.

Eventually, he was placed in the Army and would be curtail to building up the artillery used in the siege of Toulon but also leading several attacks that would decide the siege. With his fame, he would become an important person within the French Army, being an inspector of artillery and getting promoted to Colonel. However that would end with the death of Robespierre and the replacement of the Jacobi government. Napoleon was tied to Augustin Robespierre, brother of the more famous Maximilian, and so was placed in prison but thankfully for him, this new governmwnt was less bloody than the Committee.

After getting out, he was without a assignment but eventually was placed in the French topographical service. Here he was able to help plan for the future invasion of Italy and learn the finer points of strategy & logistics.

Eventually he would be called upon by the French government and Pierre Barras. There was a threat of a revolt and Napoleon was placed as Commander of the Army of the Interior and told to put down the threat anyway possible, which is the famous "whiff of grapeshot" incident.

From here, he was the Belle of the Ball, he was the Savior of the Directory, and he wanted a command. So he was given the Command of the Army of Italy, the rank of General, and would marry Josephine just before he left for Italy.

He was 28 by the time he reached full army command. The French Revolution had many commanders that jumped rank. For once, the military was home not to nobles but people who are capable. Napoleon rose because of perfect timing, connections, luck, and the change of government.