Whatever it may take to check India’s stately progress towards the semi-finals of the World Cup, they have not met it yet. They swept past South Africa and on Sunday they steamrollered Australia. They won by 36 runs but the margin was much closer than the contest, which felt, for most of the day, like it was only heading one way.

India started it with a partnership of 127 between Rohit Sharma and Shikhar Dhawan, and finished it, eight hours later, with Bhuvneshwar Kumar taking the final wicket with the last delivery. The only reason it wound up as tight as it did was because Australia fought a pigheaded rearguard in the second innings.

They were overwhelmed on the field and off it, where their solitary knot of fans in the corner of the Peter May stand looked like a pinprick of sunshine in the midst of a wide blue sky of India supporters. At one point Virat Kohli even asked his supporters to go easy. They were booing Steve Smith when he was fielding at long-off, until Kohli broke off from his batting and started shouting at them to clap him instead.

There was an uneasy moment, too, when Adam Zampa was caught on camera fiddling around with an object in his pocket while he was holding the ball. His captain, Aaron Finch, said later that Zampa was only using a handwarmer.

Kohli’s kind gesture was the only mercy he showed. Ominously, it felt in the end, he and his team still had plenty left to give if they needed it, that they had not had to bowl or bat as well as they can.

Virat Kohli catches Nathan Coulter-Nile on the boundary to end Australia’s faint hopes of victory. Photograph: Tom Jenkins/The Guardian

Their innings started slowly after Kohli won the toss. Dhawan and Sharma spent the opening overs settling in. Although, given Mitchell Starc was on at one end and Pat Cummins the other, they did not find it all that easy to get comfortable.

Starc almost had Sharma in the second over but the ball burst through Nathan Coulter-Nile’s grasp at short mid-wicket. And then Starc cracked Dhawan on the left hand with a wicked delivery that spat like hot fat from just back of a length. Dhawan needed treatment and afterwards he was careful to play him from deeper back in his crease. He did not field either.

After seven overs, India had only 22, then Finch brought on Coulter-Nile and the innings started to snowball. Dhawan clapped one four down the ground, whacked another through point, and slapped a third through third man.

India had clearly decided that if they could just see off Starc and Cummins, then the rest of Australia’s attack would be easier to get at. Finch kept making changes, fiddling around with his two spinners and bringing back Starc and Cummins when he had to but Dhawan and Sharma pressed on, accelerating steadily, up to five, then six runs an over.

Dhawan was doing most of the work. He was severe on Marcus Stoinis. He uppercut him over the wicketkeeper’s head and hit him for a preposterous inside-out cover drive. It was the 23rd over before Sharma sliced a loose drive and was caught behind. The ground fell silent but the lull did not last long because Kohli was next man in. It made for a smooth progression, although everything did seem to get a little nervous when Dhawan reached 99, the tension was increased by the heartbeat thump of all those Indian drums.

He almost blew it, too. Stoinis was bowling and Dhawan prodded the ball to mid-on. Kohli called him through but Dhawan hesitated and Kohli turned to dive back to his crease as the throw came in. The ball ricocheted off the stumps and the batsmen ran the single after all but before they or anyone else could celebrate they had to wait for the TV replay to see whether Kohli was out. It was 37th over before Australia took another wicket, when Dhawan hoicked a catch to deep mid-wicket on 117.

Shikhar Dhawan celebrates his century. Photograph: Tom Jenkins/The Guardian

Soon after, India finally slipped into something near top gear. They promoted Hardik Pandya to No 4, and he blazed 48 off 27 balls. Even his dismissal only brought in MS Dhoni, who clobbered 27 off 14. Even Starc and Cummins could not do anything much to slow them down now. Kohli finished with 82 off 77.

Facing an imposing total of 352, Australia decided to try to win it with the sort of batting tactics that were last in fashion when their coach, Justin Langer, was still playing.

David Warner played another bizarre innings of 56 from 84 balls before he was caught at deep midwicket. He is batting in shackles at the moment. Finch went well, but then he was run out hesitating over a second that Warner called for, which did not help.

Smith fared better but still was not able to score any quicker than a run a ball before he was given out lbw to Kumar for 69. Australia still needed 115 and there were only 10 overs to go. Glenn Maxwell (28 off 14 balls) and Alex Carey (55 off 35) finally started batting with some urgency but it was too little, too late.