I know that airborne troops became famous in WWII, but was that the first time they actually fought after parachuting into the combat zone?
I believe it would be Operation Weserübung, the German invasion of Denmark and Norway during the Second World War. German Fallschirmjägers were deployed in several small scale actions in both Denmark and Norway.
The first airborne assault occurred at approximately 5 a.m. on 9 April 1940, when a German battalion were dropped on the Danish island of Masnedø, where they stormed the coastal fortress and took control of the Storstrøm Bridge. Having met no resistance there, the first opposed airborne assault took place slightly later when the Germans landed at Sola Air Station and came under fire from the Norwegian base's defenders.
Perhaps the first strategically significant airborne attack occurred a month later on 10 May, when German Fallschirmjägers successfully captured the Belgium Fort Eben-Emael. The reputedly impregnable fortress guarded the country's entrance on the Albert Canal, and its capture was an important part of the Battle of Belgium.
It seems (credit to @DevSolar for finding it) Soviet paratroopers dropped at Petsamo and other points behind Finnish lines on 30 November 1939, during the Winter War. As far as I can tell they seem to have been deployed as raiders or saboteurs, but were quickly wiped out and achieved no real tactical success. I'm not sure if these were airborne assaults on military targets, or insertions of commandos behind enemy lines; if it counts however this predates the Germans by over four months.
Soviet paratroopers also fought at the earlier Battle of Khalkhin Gol in the Far East, but apparently only as ground troops.