More than a third of tests last week came back positive, the IFRC said.
"Afghanistan is at a crisis point in the battle to contain COVID-19 as hospital beds are full to capacity in the capital Kabul and in many areas," said Nilab Mobarez, Acting President of the Afghan Red Crescent Society, in a statement released by the International Federation of Red Cross and Red Crescent Societies.
The surge was putting intense strain on a country where millions already live in poverty and health resources are scarce.
Health authorities on Thursday registered 2,313 positive cases and a record 101 deaths from COVID-19 in the last 24 hours. Officials and experts have said low testing means those official figures are probably a dramatic undercount.
Afghanistan's fragile health system has been damaged by decades of war. Violence has risen in recent months, with US-led foreign forces withdrawing by September and peace negotiations between the Afghan government and insurgent Taliban largely stalled.
Major hospitals have closed their doors this week to new COVID-19 patients after an influx of cases left them with a lack of beds and oxygen shortages.
The IFRC warned that lack of vaccine access and hesitancy were exacerbating the situation. Less than 0.5 percent of Afghans have been fully vaccinated.
Around 700,000 doses of China's Sinopharm vaccine arrived in the country last week, allowing authorities to start the next round of its vaccination campaign.
The IFRC was working with Afghan authorities to provide more resources and try and boost medical oxygen production, according to Necephor Mghendi, the head of Afghanistan Country Delegation for IFRC.
"More international support is needed to help win this race against this virus, so we can save thousands of lives," he said.