The midfielder joins on loan from League Two Bury on is 21st birthday.
The Red Dragons lead the early season table and manager Sam Ricketts thinks Burgess’ arrival will only increase their chances of promotion.
The deal will see him stay at the Racecourse for six months.
Ricketts told the club’s website: “Scott is a young lad who knows the National League very well. He was promoted with Macclesfield last season so he knows what it takes to be successful.
“I’m looking forward to working with him here at Wrexham and I hope he enjoys his time with us.”
Burgess added: “I’m really excited about joining Wrexham. The team has had an amazing start to the campaign and hopefully I am able to come in and help out with that.
“I was with Macclesfield Town last season and we went up as champions, so I know all about the National League.
“I’m here at Wrexham now and the aim is to do the same this season.”