Mercedes driver George Russell, who won at Melbourne last week, took out the sprint from Charles Leclerc and Lewis Hamilton.
Lawson started 13th on the grid, but gradually made ground during the 19-lap dash, moving into fifth with five laps to go, before losing a couple of places in the final two laps.
Driving on hard tyres, Lawson stayed out on the track, while others opted to pit.
His two points were his first this Formula 1 season, after finishing 13th at the Australian Grand Prix.
Lawson's Racing Bulls team hailed his drive, but the Kiwi said the decision to use up a set of hard tyres in the sprint takes away an option for the Grand Prix.
"That is the problem," he told Sky Sport. "We are one hard down and I think when we made the decision this morning we had to think about our speed.
"We tried to maximise today, take the advantage. Tomorrow will be hard for us to fight for the points.
"We need to find some speed, which we will try to do this afternoon [in qualifying], but yes, it potentially hurts us a little bit."
Russell held off the fast-starting Ferraris to stretch his championship lead to 11 points.
The Briton started on pole position at the Shanghai International Circuit and finished ahead of Ferrari duo Leclerc and Hamilton, after a late safety-car period.
Leclerc finished 0.674s behind Russell, after an earlier battle with Hamilton, as Ferrari had two cars in the top three of a Formula 1 race of any sort for the first time since 2024.
Hamilton and Russell battled for the lead early, with four changes in the opening five laps, before the Mercedes driver made a move that stuck and secured the eight points available for a sprint win.
Reigning champion Lando Norris finished fourth, with Mercedes' Kimi Antonelli fifth, after serving a 10-second penalty for a clash with Red Bull's Isack Hadjar on the opening lap.
Oscar Piastri finished sixth for McLaren, with Lawson seventh and Oliver Bearman taking the final point for Haas.
After the later qualifying sessions, Lawson will start from 14th on the grid in the main race.
With the bottom six drivers dropping out after the first qualifying session, Lawson scraped through into the second stage in last place.
He was eliminated, when a double yellow flag denied his last chance at bettering his lap time.
Kimi Antonelli will start from the pole, with Mercedes teammate George Russell alongside, Lewis Hamilton and Charles Leclerc on the second row for Ferrari, and McLaren stablemates Oscar Piastri and Lando Norris on the third line.
- Additional reporting RNZ
