Piastri sealed his win with a trademark committed, decisive move on Norris when the conditions were at their most treacherous when the race finally started.
The original start was abandoned because of heavy rain and poor visibility after the formation lap behind the safety car.
The drivers then sat in the pit lane for an hour and 20 minutes, followed by four laps behind the safety car before the race was finally allowed to start 90 minutes later than scheduled.
Piastri tracked Norris closely through the first corner and through the high-speed swerves at Eau Rouge before diving around the outside into the Les Combes chicane at the end of the long Kemmel straight.
Norris complained over the radio that he was down on battery power, but was told he had used it up at the start behind the safety car.
And after the race he said that Piastri had simply done a better first lap by pushing harder through Eau Rouge, where in the wet drivers have to choose how much to lift off, when it is flat in the dry.
“Oscar did a good job, nothing more to say,” Norris said. “Committed a bit more through Eau Rouge and had the slipstream and got the run and that was it. Love to be up top but Oscar deserved it today.”
Piastri said: “I knew that lap one was going to be probably my best chance of winning the race.
“I got a good exit out of Turn One and then lifted as little as I dared through Eau Rouge and it worked out pretty well. We had it mostly under control after that.
“I was a bit disappointed it was a rolling start because I thought that would take away some opportunity but when I was that close I knew I was going to lift a little bit less than Lando did. A bit lively over the hill but then the slipstream helped me out.”
Once in front, Piastri inched away in the lead until he was just under two seconds in front when he chose to stop for slick tyres on lap 12, his position in front giving him priority on stop timing and forcing Norris into a difficult position.
Piastri fitted the medium tyres while Norris had to do an extra lap on a drying track on highly worn intermediate tyres.
His engineer asked him if he would like hard tyres and try to run to the end, a decision Norris agreed with, and he rejoined 9.1 seconds back from Piastri after his stop.
By around lap 20, Piastri told his engineer that he thought it would be “tough” to get his intermediates to the end but for a long time he held the lead at about eight seconds, and it slowly became apparent he had decided not to stop again.
In the final few laps, Norris began to make significant inroads into Piastri’s lead, and was within four seconds of the leader with three laps to go.
Norris, who said he drove the entire stint as if he was on qualifying laps, cost himself about four seconds in his chase, with three errors – running wide at the fast Pouhon left-hander at one point, and a couple of lock-ups at La Source. He also had a slower pit stop than Piastri when the mechanics struggled to fit his left-front wheel.
But Piastri managed the gap expertly to win by 3.4 seconds.
Behind the leaders, the drivers were stuck in their positions after the pit stops, even if there was some tension for Leclerc as Verstappen pushed him hard in the closing laps.
Much of the excitement in the race was provided by Hamilton. The seven-time champion started from the pit lane after Ferrari decided to change his set-up after his error in exceeding track limits in qualifying left him down in 16th on the grid.
And Hamilton justified the decision with a series of excellent, improvisational overtaking moves to rise to 13th place before becoming the first driver to stop for slicks on lap 11, one before Piastri.
That won him a chunk more places, and Hamilton was promoted to seventh by the pit-stop period, which he held to the end of the race.
He closed to within a second of Albon on the final lap but was unable to pass.