Nikola Vujadinovic then converted to put Sturm 1-0 head suggesting the visitors were in for a long, hot afternoon.
Instead, Rapid turned the game around with second half goals from Christopher Trimmel, Guido Burgstaller, Brian Behrendt and Louis Schaub, while Daniel Beichler replied for the hosts.
The match completed a dramatic weekend in the Austrian Bundesliga, where a heatwave forced most games to feature water breaks midway through each half.
On Saturday, Admira Wacker had three players sent off, two of them in the first half hour, as they crashed 7-1 at promoted Groedig despite taking a sixth-minute lead through Stefan Schwab.
Admira, coached by Austria's leading scorer at international level Toni Polster, were left bottom of the table after three straight defeats.
Champions Austria Vienna led 2-0 and 3-1 against SV Ried before the visitors scored twice in the last 10 minutes to draw 3-3 and title favourites Salzburg were held 1-1 at Wacker Innsbruck.
(Reporting By Brian Homewood, editing by Pritha Sarkar)
