The Sri Lankans, who were chasing 240 for victory, looked dead and buried at 107 for eight before Mathews, who ended up with an unbeaten 77, and Malinga lit up the Melbourne night with some batting pyrotechnics to claim an unlikely victory.
The partnership was finally broken when Malinga was run out for 56 with the scores level at 239, leaving 10th man Muttiah Muralitharan to hit the winning boundary.
“I’m still catching my breath, it was awesome,” Mathews told reporters. “All credit should go to Lasith, it was very easy batting with him.
“What a performance, we had to fight for every run ... there’s nothing I can say but Angelo Mathews and Lasith Malinga, unbelievable.”
Australia have not tasted victory in nine matches in all forms of cricket going back to July and have the Ashes series against England looming at the end of the month.
“I honestly don’t know how we lost that game,” said Michael Clarke, who was captain in place of Ricky Ponting who missed the match to attend his grandmother’s funeral.
“Very disappointing ... just seems to be right at the end when we get close to winning matches, we just can’t seem to get that win over the line.”
It had looked like being left arm spinner Xavier Doherty’s night as he took four wickets and engineered a stunning run out on his debut to put the hosts in the driving seat.
Mathews and Malinga had other thoughts, however, and they sprayed the ball around the Melbourne Cricket Ground with abandon to exceed Kapil Dev and Syed Kirmani’s ninth wicket stand of 126 for India against Zimbabwe in 1983.
Sri Lankan paceman Thisara Perera earlier took five for 46 but Michael Hussey belied his recent poor form with an unbeaten 71 as Australia rallied to 239-8.
The Australians, beaten by seven wickets in a Twenty20 match against the Sri Lankans in Perth on Sunday, were cruising at 85-1 until Perera claimed three wickets in eight deliveries.
Clarke (27) and wicketkeeper Brad Haddin (49) went to superb catches from Kumar Sangakkara and Tillakaratne Dilshan before Cameron White was skittled for a golden duck.
Doherty enjoyed his own eight-delivery spell to rival Perera’s, first trapping Mahela Jayawardene leg before in his first over and having Chamara Silva caught by Shane Watson and bowling Sangakkara in successive balls in his second.
The second match is in Sydney on Friday and the series closes with a third day-night match in Brisbane on Sunday. The first Ashes test begins in Brisbane on Nov. 25.
