The Wearside outfit were unable to take advantage of Fabian Delph's dismissal early in the second half and had to settle for a point at Villa Park.
Delph was shown a straight red card for a late challenge on Jordi Gomez, but Sunderland could not find a way past Brad Guzan as Villa stood firm.
Guzan produced a fingertip save to keep out Gomez's late strike, but Villa also had chances to snatch all three points.
Sunderland head coach Poyet was left to rue his side's inability to come up with a winner in their final game of the year.
He said: "It feels very much like we’ve lost two points today as the game was there to take.
"We tried, there's nobody can say that we didn't try as we were constantly pushing forward, but we just didn't find the back of the net.
"We were just missing a quality moment - something similar to what Adam Johnson has done for is in the last two games - but it just didn’t happen for us,
"Jordi went close right at the end, but at the moment it's just not going for us. We must keep believing and keep improving on what we do."
He added: "We should have made more of the extra man but we didn't. The effort was there but we just couldn't manage to make more of it."
Sunderland face a trip to champions Manchester City on New Year's Day.
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