Goals from Connor Wickham and Jermain Defoe ended a run of five league games without a win for Poyet's side and put them four points clear of the relegation zone.
The Uruguayan is now looking for his squad to build on the win in a run of top-flight matches that sees them face Swansea City, QPR and West Brom after taking on Fulham in the FA Cup on Tuesday.
"It was much needed," he told BBC Sport. "We knew Burnley would be hard-working. First and foremost we needed to match that, and we did. Then we needed to impose ourselves with quality.
"The players on the pitch knew how important it was. We don't need to think we are okay - we need to go to the next level.
"I knew that Jermain would score sooner or later because his qualities are there. I want this to be a turning point. I hope we will go into the next game with the confidence we were looking for."
The win is only Sunderland's second at home this season in the Premier League, and Defoe knows the importance of his first strike since arriving from Toronto FC earlier this month.
"It was a massive win," said Defoe. "We managed to get the early goal which helped us, settled us down.
"I have a mixture of happiness and relief. We needed a performance and a victory like today, so it was very welcome.
"We knew Burnley and the situation we were in, so we needed to be very good with commitment, desire and a fighting spirit.
"We kept a clean sheet and the two strikers who started today scored, so it's a perfect result for us."
