The 1-0 Boxing Day triumph at fifth-placed Everton was not the first notable top-flight victory the Black Cats have achieved since they appointed Poyet in October.
Others have come against Newcastle (sixth at present) and Manchester City (second), while in the Capital One Cup, defeats of Southampton (ninth in the league) and Chelsea (third) have also shown the Wearsiders' capacity under Poyet for seeing off decent outfits.
But the first three wins on that list are the only ones they have managed in the Premier League all season, and they find themselves currently bottom of the division, three points adrift of safety.
And Poyet, while somewhat baffled, is in no doubt that Sunderland's failure to beat teams who are closer to them in the table is something which needs to be addressed if they are going to stay up this term.
Asked about the wins his side have registered and the contrast to their other results, he said: "I have no explanation for that.
"I don't know how or why. But it is us - we cannot blame anybody else, because we have had chances to beat the teams around us.
"Always we do something wrong. Then, when we play against the top teams, something happens the other way.
"We need to change it if we want to stay up."
A 25th-minute penalty from on-loan Swansea midfielder Ki Sung-yueng helped steer Sunderland at Goodison Park to what was their first victory in seven league matches, and maiden away win of the season.
The spot-kick was awarded after Leon Osman miscontrolled a goal-kick from Everton goalkeeper Tim Howard.
Ki latched onto the ball and darted into the box, where he hit the turf when tackled by Howard, who was shown a straight red card.
The Toffees piled on the pressure after the break but could not find an equaliser, with Black Cats stopper Vito Mannone - who Poyet said "probably won the game" for the visitors - pulling off several impressive saves.
Reflecting on his team's second league loss of the campaign, and first at home in the top flight in 2013, Everton boss Roberto Martinez said of the build-up to the goal: "Both players (Howard and Osman) probably got their decisions wrong. But we are going to learn from this.
"The referee was spot on (in awarding the penalty) - I think he made the correct decision.
"I couldn't be prouder of the second-half performance. I thought we deserved an equaliser.
"I think our performance in the second half was one of our strongest and we've learned a lot about ourselves.
"For me, although we don't have a point from this, we have a stronger team going into the second half of the season."