In front of a packed stadium with 20,200 spectators watching on, the 2016 Winter Youth Olympic Games silver medallist jumped 133.0 and 120.5 metres, the same distances as runner-up Dawid Kubacki of Poland.
Lindvik was awarded 253.3 points for his jumps, outscoring Kubacki by just 1.3 points.
The Norwegian's teammate Daniel-André Tande came third, four points behind Lindvik. Tande made up for a disappointing 126.0-metre first jump by leaping 131.0 metres in the second round, comfortably the furthest jump in that round amidst changing wind conditions for the athletes.
Japan's Ryoyu Kobayashi, the defending Four Hills Tournament champion and winner of five straight Four Hills events until Lindvik's win at Garmisch-Partenkirchen on New Year's Day, struggled with the weather conditions and could only put together jumps of 122.0 and 120.0 metres.
The 23-year-old's efforts were good for 229.8 points, placing him down in 14th in the Austrian city. He finished ahead of Olympic champion and two-time Four Hills overall winner Kamil Stoch, who was 15th.
Four Hills overall leader Kobayashi came into the event with a 6.3-point lead over Germany's Karl Geiger, who finished eighth in Innsbruck. Geiger scored 236.5 points (jumps of 117.5 and 126.0 metres), which was enough to overturn that lead and put him ahead of the Japanese by 0.4 points.
However, neither man's results were enough to keep them in the top two of the overall standings.
Kubacki's big score sent him into the lead on 830.7 points, while Lindvik is on his tail 9.1 points behind. Geiger and Kobayashi sit third and fourth respectively, with only 13.7 points separating Kubacki from Kobayashi.
The Four Hills Tournament concludes on Monday (6 January) in Bischofshofen with the tournament still wide open.