Councillors alerted residents about the issues when the service was delayed because of the number of workers absent because of the virus yesterday, Tuesday, June 29.
Staff were moved in to keep collections going, with the only two streets left without a visit due to be checked early today.
It comes as the number of people testing positive for Covid-19 in the region rises.
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The council’s deputy leader Councillor Claire Rowntree said: “Covid case numbers are continuing to rise rapidly in Sunderland, in line with the situation regionally and nationally.
"This is reflected in our own workforce with a number of staff away from work and self-isolating due to Covid.
"We have tried our very hardest to ensure that despite this, refuse collections for our residents could operate as normal as possible.
"Due to a number of staff being off as a result of Covid some refuse collection rounds in the city started a little later than usual on Tuesday as new staff were brought in and provided with all required training and health and safety inductions.
“All collections were completed across the city apart from a two streets in Red House that were due to be collected first thing Wednesday morning.
"This has been a fantastic effort by all of the Refuse Team in responding to an unprecedented service impact due to Covid.
"We will continue to monitor and respond to the Covid impact so that residents receive services they expect.”
Silksworth Councillor Phil Tye sent a message to his ward about the issues, with residents appreciating the notice.
“I’m really pleased that the council were able to divert resources from other areas to offer cover,” he said.
"I think it’s important because residents understand that sometimes there are operational issues, but it’s OK if they know about them.
"This is really the first time we knew about it before, rather than about changes at the end, and it meant we were able to communicate that to residents.”