Schwimmer will join the group on 1 August, succeeding Xavier Rolet, who left amid a bitter public row with an investor to oust the group's chairman, which was sparked by the announcement in November 2017 that Rolet would step down as chief executive.
The board of the FTSE 100 firm eventually beat off the attempt by Sir Chris Hohn to remove Donald Brydon as chairman.
LSE said David Warren, who stepped in as interim chief exec, will return to his role as chief financial officer.
Most recently, Schwimmer was global head of market structure and global head of metals and mining in investment banking at Goldman Sachs. His career at the US banking giant has spanned 20 years.
Schwimmer also worked as chief of staff to then-president and COO (now chief executive) Lloyd Blankfein and spent three years in Moscow as co-head of Goldman Sachs' business for Russia.
Donald Brydon, chairman of the LSE, said:
David is a leader with great experience in the financial market infrastructure sector, which he has been closely involved in throughout his investment banking career, as well as capital markets experience in both developed and emerging markets.
He is well known for his robust intellect and partnership approach with clients and colleagues alike.
Schwimmer said it was an "honour and privilege" to be asked to lead the group.
"It is both an iconic institution and a great business. Having worked with exchanges and other market infrastructure companies for much of the past 20 years, I have been impressed by its strong track record of partnering with customers to deliver innovative solutions," he said.
Schwimmer said the group has "multiple opportunities for further attractive growth" across its capital formation, information services and post trade businesses.
"I look forward to working alongside the group's highly capable management team to continue to deliver value for its customers, employees and shareholders," he said.
During Rolet's tenure, the LSE's market capitalisation grew from £800m to £14bn, and last month it revealed profit had jumped in 2017 as it recorded growth across all of its core businesses.