The Gunners did not really get going until the 60th minute when Bukayo Saka celebrated his first England call-up with a header from Hector Bellerin's cross.
Three minutes later they doubled their lead with Nicolas Pepe's good run and finish into the bottom corner - finishing off a move with 19 passes.
They looked completely in control until David McGoldrick curled home the Blades' first league goal of the season.
But it was not enough to affect the outcome and United have now lost seven consecutive top-flight games for the first time since 1975.
Saka said afterwards: "All the things that have happened to me. I feel blessed. I'm a happy man.
"I don't get a lot of headers but it fell perfectly for me. I told Hector what a cross it was. It is the kind of patterns we work on.
"I am trying to do my best when the boss picks me. I try to repay the trust.
"We gave so much credit to Sheffield United as they are a good side. We are happy to get the points."