The 25-year-old was forced off with the hamstring problem in the first half of Sunday's match, but a scan has revealed that he could return in under a month.
Simpson told the club's official website: "It's a lot better than I first thought. I have never had a hamstring problem before. I just felt something bad and I knew straight away I would have to come off.
"But the scan results came back on Wednesday and it is hopefully three weeks, maybe four maximum."
The full-back has made 116 appearances for Newcastle United since he joined the club from Manchester United in 2009.