Sanitary conditions inside the stadium are said to be poor with only 10 to 15 stalls of latrines for tens of thousands of people. Many families are being forced to camp in open fields nearby, according to the agency.
"Numbers keep swelling, it's a race against time to try and save lives," said Kasina, adding that the needs are fast outpacing the aid being provided.
The refugees include a large number of unaccompanied children separated from their families, the agency says.
On February 21, UNHCR told a press briefing in Geneva that it would seek to move people from the stadium. However logistical challenges mean it takes six to eight hours to move large numbers of people to the Musenyi refugee site in southern Burundi. That site, which can host 10,000 people, is now 60% full, according to the agency.
The agency has urged countries to contribute to its emergency appeal for $40.4m (R738.9m) for lifesaving help to support the potential influx of 258,000 refugees into Burundi, Tanzania and Zambia.