Former Manchester United full-back Phil Bardsley's 21st-minute strike - his second goal in five days - proved enough to claim three priceless points for the Black Cats after he shrugged off James Milner and curled a shot across keeper Costel Pantilimon and inside the far post.
Asked if that was the biggest victory of his managerial career to date, even eclipsing a 2-1 defeat of arch-rivals Newcastle, he said: "Yes, probably.
"I know that for the fans, for the club, for the history, it's always more important to beat the biggest rivals without any doubt. That's when you have to win.
"But in terms of opposition, quality, world-class players, this is the biggest of my career, so it's going to be a good evening without any doubt.
"I don't know how my heart is right now. I don't want to check it just in case it's bad news, but it's going to be a great evening.
"I am so pleased for the players. They need to believe. It's easy to talk - we can say things and you can write. But at the end of the day, it's the players on the pitch who need to be convinced that this new way of playing football and defending is working, which is great for the future.
"So let's see if we can keep this going. It's a good time to go into the international break."
Sunderland remain three points adrift of safety, but having won their last three home games, two of them in the league, the table makes for much more satisfying reading.
Poyet said: "Right now, yes. Three hours ago? I'll tell you what, when I looked at that table, it was awful - and we were playing Manchester City.
"But that's the key. It's the feeling that you are sending to the rest above you. I would be thinking, 'They are coming'."
- Sports & Recreation
- Phil Bardsley
- James Milner
- Manchester United
- Costel Pantilimon