Sunderland dragged themselves back into the match within seconds through Fabio Borini's effort, but they were second-best for an hour.
However, the Saints failed to capitalise on the wealth of opportunities they created and they were made to pay with 19 minutes remaining when Adam Johnson smashed a stinging shot through keeper Artur Boruc's hands to snatch a point.
Poyet was relieved to have emerged with a point after a dreadful start to the game, and is hoping his players will head into Wednesday night's decisive Capital One Cup semi-final clash with Manchester United free of pressure as a result.
He said: "It's a very good point. I am very realistic, it looked bad, really bad. It looked really dark.
"It looked like nothing was right for us, so at that time if you had said to me, 'You are going to get a point from the game' I would have taken it, for sure.
"In the end, we got a point - let's be happy and let's make sure we go on Wednesday with a different feeling to enjoy Old Trafford and try to do something special, but without the pressure of the points in the Barclays Premier League."
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball