The Premier League side ultimately saw off the Championship strugglers thanks to Patrick van Aanholt's first-half strike and, although Leeds were a threat after the interval, Sunderland should have found the net on multiple occasions.
Santiago Vergini and Steven Fletcher both went close before the break, while Ricky Alvarez struck the crossbar in the 13th minute.
And Poyet, whose side also spurned opportunities in the second half, was disappointed at failing to make the final scoreline more comfortable.
"We needed to kill the game after being the better team in the first half and creating so many chances," the head coach told Sunderland's official website.
"It wasn't necessarily down to Leeds' goalkeeper that we didn't score, but perhaps our finishing. We needed to be more clinical.
"I think we allowed Leeds to get back into the game. We could have had it finished before half-time, but they came back out for the second half and threw everything at us.
"However, it was a cup game and we needed to get through and win and we did that – we're in the draw for the next round."
One thing Poyet was delighted about, though, was the return of scorer Van Aanholt, who was playing his first game since picking up a shoulder injury at the start of November.
"It was a big decision to include Patrick," the Uruguayan added. "It was tough, especially with our full-back situation.
"With all due respect to Leeds, we told Patrick to treat the game as a friendly and for him to ease his way back into it, as you do when you're returning from an injury to make sure that you're okay.
"It was good for him and for us it was perfect, he played a full 90 minutes and scored a goal and got through it all without any problems.
"We know what qualities he brings to the side and we've not seen him score before – he's even scored before Santi [Vergini], which is surprising when you look at some of the chances he's had in the last year."
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