Liverpool took the lead in the 28th minute when Sturridge tried to head home a Steven Gerrard corner but ended up glancing it in with his arm.
Sturridge later set up two goals for Suarez as the Uruguayan made his first Premier League appearance since April, although Liverpool were made to work hard by a resilient Sunderland side.
"It was very hard," Sturridge told Sky Sports. "I don't think we played as well as we could have in the first half but we were 2-0 up, and then I think we took our foot off the pedal a little bit in the second half."
Asked about his goal, Sturridge said: "It was a great ball in from Stevie and I've obviously tried to head it, but I guess my heading's not good enough is it?
"I've missed the ball and it's gone in off my arm but I wouldn't do that on purpose and sometimes these things happen."
Sturridge was pleased to play alongside Suarez once again and hopes they can continue to link up to great effect in the future.
He said: "It's great to have him back and it gives us options. He is one of the best forwards in the league and we form a good partnership."
Liverpool have enjoyed a consistent start to the season and the result lifted them to second in the standings, but Sturridge refused to get too carried away.
Questioned on whether they were genuine title contenders, he said: "It is early days. We will take every game as it comes. That is the most important thing, worrying about the here and now and not the future."