India needed their sub-continental rivals to lose the last round-robin clash to keep their tour alive but Sri Lanka made 238 all out from their 50 overs before dismissing Australia, who had already secured their place in the final, for 229.
Australia went into the final over of their innings needing 10 runs for victory but Nuwan Kulasekera had David Hussey caught for 74 with the first ball to spark emotional celebrations among the Sri Lankans.
All-rounder Christian had earlier taken the 31st hat-trick in one-day internationals – just the fourth by an Australian – by removing Thisara Perera, Sachithra Senanayake and Kulasekera in successive balls in Sri Lanka’s 44th over.
The 28-year-old, who made his international debut earlier this month, had Perera caught for five at deep midwicket by Mike Hussey and then trapped Senanayake and Kulasekera, more contentiously, lbw to prompt wild scenes from his team mates.
Sri Lanka had recovered from a faltering start when Kumar Sangakkara (64) and Dinesh Chandimal (75) put on 123 for the third wicket.
Quick James Pattinson (4-51) removed them both and later dismissed Lahiru Thirimanne (51) to celebrate his return to international cricket as Sri Lanka stumbled in the last 10 overs.
Christian bowled Lasith Malinga with the final delivery of the innings for an even more impressive return of five for 31 from his nine overs – his first international five-wicket haul.
Australia also made a stuttering start to their innings and lost their first three batsmen for 26 runs before stand-in captain Shane Watson and Mike Hussey (29) combined for 87 to steady the ship.
However, Watson, who was looking well set on 65, then had his stumps shattered by a brilliant yorker from Malinga (4-49).
Sri Lanka sensed their chance and, cheered on by the majority of fans in the 30,000 crowd at the Melbourne Cricket Ground, seized it.
While David Hussey – who hit his fifth half century of the series – was around the result was still in doubt but when he sent the first ball of the final over towards the safe hands of Tillakaratne Dilshan at long off, Sri Lanka were off to Adelaide for Sunday’s first match of the final.