The reigning ICC Test player of the year retired hurt on 27 after a delivery from paceman Mitchell Johnson hit him on the left index finger on the third and final day of Sri Lanka's humiliating defeat, which saw them go down by an innings and 201 runs.
The left-hander immediately left the field with excessive swelling to his hand and was taken to hospital for scans which revealed a compound fracture.
Sangakkara is now set to miss the third and final Test, where Sri Lanka will attempt to avoid a 3-0 series whitewash, the five-match one-day international series and the two Twenty20 clashes which wrap up the tour.
Sri Lanka were also without Prasanna Jayawardene (broken thumb) and left-arm quick Chanaka Welegedara (hamstring) for their second innings in Melbourne, where the tourists were skittled for just 103.
Sri Lanka captain Mahela Jayawardene admitted the injuries were frustrating but could not be used to excuse his team's performance.
"These are things that we can't control, (but) they are not excuses. Still with those guys playing we probably would have got beaten," he said.
"So we just need to refocus our thoughts and come back strongly.
"We started the tour off pretty well in Hobart with the Test match but here we seemed to never get anything going for us. We just gave up."
Victorian all-rounder Glenn Maxwell is in line for a Test debut, meanwhile, after being called up to replace Australia's injured vice-captain Shane Watson for next week's third Test in Sydney.
Watson aggravated a chronic calf injury on his way to making 83 during Australia's crushing Boxing Day Test victory at the MCG. Skipper Michael Clarke told reporters after the match that the injury-plagued all-rounder will miss at least the third Test, which starts on Thursday at the SCG.
Maxwell, renowned for his explosive hitting and sharp fielding, boasts a batting average of 42 and bowling average of 33.38 after two years of first-class experience with the Bushrangers.
The 24-year-old off-spinner could win himself selection for Australia's tour of India early next year if he performs productively with the ball in spin-friendly Sydney.
Still battling hamstring soreness that threatened to keep him out of the Melbourne Test, Clarke himself is no certainty to play despite being the match's only centurion as Australia beat Sri Lanka by an innings and 201 runs.
Queensland batsman Usman Khawaja has been placed on standby.
Left-arm paceman Mitchell Starc will return to the Test side after being controversially asked to carry the drinks in Melbourne.
And given it will be highly unlikely that four quicks will be taken into the Sydney Test, incumbent pacemen Mitchell Johnson, Peter Siddle and Jackson Bird will most likely be fighting for the other two pace bowling spots.
Australia enter the New Year Test with an unassailable 2-0 series lead.
Australia's 13-man squad: Michael Clarke (capt), Jackson Bird, Ed Cowan, Phil Hughes, Mike Hussey, Mitchell Johnson, Usman Khawaja, Nathan Lyon, Peter Siddle, Matthew Wade, David Warner, Mitchell Starc, Glenn Maxwell.