The Italy international somehow managed to scoop the ball over the crossbar from two yards with the goal gaping after being picked out by Phil Bardsley's knock down on the stroke of half-time.
It left Sunderland without a goal in their last five away matches in the Premier League but they did climb off the foot of the table.
After the game, which ended the visitors' run of five consecutive away league defeats, Poyet shrugged off the miss as an occupational hazard.
"He's so sad," he said. "He doesn't stop thinking about it and saying sorry but that's football and things happen for a reason sometimes."
Poyet went on to underline the positives his side can take from the result.
"For us it was our best performance away from home because we made it difficult for them," he said.
"We were brave on the ball, we created two or three very good chances so for a team down at the bottom playing away from home I don't think it's too bad.
"Maybe we deserved three points today but I want that team on the pitch every week and if it's going to be like that we'll win plenty of games."
The Uruguayan defended his side's gritty performance although they did come closest to breaking the deadlock on two occasions, with Giaccherini's miss and Fabio Borini's second-half header, which crashed off the bar.
"There's always been something missing but today overall we defended well and with the ball we played our way," he said.
"We need to play and try to maintain the pace of the game, sometimes it's slow but we'll play quicker when we get better.
"It's our way of playing, I don't think you can say it's good or bad it's just a way but for us it's very important to control the game if we can.
"We're off the bottom, a clean sheet away from home, with no own goals, no sendings off, no mistakes.
"I asked for that before the game, I said, 'give me a normal game away from home.'"
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