With captain Eoin Morgan not playing due to a fractured finger, England had another scare when fast bowlers Mark Wood and Jofra Archer sustained injuries.
Wood was troubled by his left foot and bowled only three overs before leaving the field for treatment while Archer injured himself while sliding to save a boundary before returning 30 minutes later.
"Mark Wood is to have a precautionary scan," the England Cricket Board said in a statement, as reported by ESPN.
"England's medical team want to ensure there is nothing serious in the build-up to the start of the World Cup."
Smith hit eight fours and three sixes in his 116 off 102 balls before he was caught and bowled by Tom Curran in the final over.
"To get 300 was superb," Australian skipper Aaron Finch said. "Steve played an unbelievable knock."
England's Liam Plunkett conceded 69 runs in his nine overs, but picked up four wickets -- including top-order batsmen David Warner (43) and Shaun Marsh (30).
"The pitch was slower than some of the pitches we've played on before," England's acting captain Jos Buttler said. "It was still a good wicket, but didn't have the pace it normally has."
England's top scorer was James Vince, who hit a patient 64 off 76 deliveries, before he was caught by Nathan Lyon attempting to cut a short ball. Buttler chipped in with a quick-fire 52 off 31 balls which included three sixes.
England play one more warm-up match against Afghanistan on Monday before kicking off their World Cup campaign against South Africa on Thursday.
Australia play Sri Lanka on Monday before their first match of the campaign on June 1 against Afghanistan.
(Reporting by Rohith Nair in Bengaluru, editing by Ed Osmond)
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