The Gunners took the lead through Santi Cazorla, only for defender Carl Jenkinson to see red shortly after the hour mark.
|Evens||Arsenal are evens with BetVictor to finish in the top four
"I thought we deserved the win and the urgency in our own penalty area was fantastic," Szczesny told the club’s official website.
"Every ball that was played across, we were the first to clear it, so they [the defence] deserve a lot of credit for the way we played.
"I think we proved to a lot of people that we have what it takes to be successful because it wasn't the best game in terms of quality, but we showed everyone the character and desire we have to get a result.
"We are delighted with the way we won the game. You always expect a very tough game when you go up north; they play very direct football and are very strong. They had a real desire to get a result and I thought Sunderland played really well, but we stood up against it and showed character."
Szczesny was also full of praise for Bacary Sagna who played as an emergency centre-back and could be forced into playing as a makeshift left back in the Champions League against Bayern Munich with new signing Nacho Monreal ineligible and other injury woes in defence.
"Bacary [Sagna] came in at centre back and cleared everything that came into the box," Szczesny added.
"It was difficult once Carl got sent off, and we were playing with a back four that had never played together. But I thought they all did a tremendous job."