Leg-spinner Leie, 28, and Phangsio took three wickets each as Bangladesh were bundled out for 138 in the final over of their innings, replying to South Africa's 169 for four wickets.
Leie dismissed Soumya Sarkar (37) just when the batsman had started looking ominous and also sent back Sabbir Rahman and the set Mushfiqur Rahim off successive deliveries to end Bangladesh's hopes.
He was adjudged man of the match for his figures of 3-16 in his international debut.
"I am very pleased, it was a very good start. I just thought I will keep it simple and hit my lengths," Leie said at the presentation ceremony.
Fast bowler Kyle Abbott (3-20) polished off the Bangladesh tail, clean bowling all three victims.
Sarkar and opener Tamim Iqbal gave Bangladesh, who lost the first match by 52 runs on Sunday, hope with a quickfire opening stand of 46 but the team kept losing wickets from there on.
Sarkar hit six boundaries and a six in his 21-ball knock before being out stumped.
Earlier, an opening stand of 95 between Quinton de Kock (44)and AB de Villiers (40) and a brisk fifth-wicket partnership at the end helped set up South Africa's total.
After the opening partnership, South Africa wobbled losing three wickets for seven runs but David Miller (30) and Rilee Rossouw (19) remained unbeaten after adding 33 runs in 14 balls to prop up the innings.
The teams will now play three ODIs, starting on Friday, before the two-Test series begins in Chittagong from July 21.
Scorecard
South Africa innings
AB de Villiers c Mortaza b Sunny 2
Quinton de Kock c Litton Das b Nasir Hossain 12
Faf Du Plessis not out 79
JP Duminy c Nasir Hossain b Sunny 18
David Miller lbw Shakib Al Hasan 1
Rilee Rossouw not out 31
Extras: (2lb, 2w, 1nb) 5
Total: (For 4 wickets) 148 Overs: 20.
Fall of wickets: 1-2, 2-31, 3-77, 4-90.
Bowling: Arafat Sunny 3-0-19-2, Nasir Hossain 4-0-37-1 (2w), Shakib Al Hasan 4-0-24-1, Mustafizur Rahman 4-0-30-0 (1nb), Sohag Gazi 2-0-16-0, Mashafe Bin Mortaza 3-0-20-0.
Bangladesh innings
Tamim Iqbal c de Kock b Abbott 5
Soumya Sarkar c Duminy b Rabada 7
Shakib Al Hasan c Parnell b Wiese 26
Mushfiqur Rahim c Miller b Duminy 17
Shabbir Rahman c de Kock b Duminy 4
Nasir Hossain c Rossouw b Phangiso 1
Litton Das c de Villiers Wiese 22
Mashrafe Bin Mortaza c de Villiers b Rabada 5
Sohag Gazi run out 3
Arafat Sunny not out 1
Mustafizur Rahman b Parnell 1
Extras: (2lb, 2w) 4
Total: (all out) 96 Overs: 18.5.
Fall of wickets: 1-7, 2-13, 3-50, 4-56, 5-57, 6-71, 7-86, 8-93, 9-94.
Bowling: Kyle Abbott 2-0-10-1 (1w), Kagiso Rabada 3-0-28-2, Wayne Parnell 2.5-0-12-1, David Wiese 3-0-12-2 (1w), Aaron Phangiso 4-0-21-1, JP Duminy 4-0-11-2.
Image: South African players celebrate after picking up a wicket
Photograph: Hannah Peters/Getty Images