Greenback, 16, who was part of the 4x100m freestyle medley relay gold medal winning team on Tuesday, finished in one minute 59.03 seconds.
China's Guangyuan Li (1:56:94) claimed gold, with Russia's Evgeny Rylov (1:57:08) second.
"I'm delighted to be leaving the Youth Olympic Games with an individual medal," he said.
"It's been a long season and I am exhausted now, but this makes it all worth it."
His success means Team GB have now won 14 medals in China with five days of competition still remaining.
Great Britain at the 2014 Youth Olympics
Ben Dijkstra (2) (triathlon; multi-nation mixed relay triathlon)
Jessica Fullalove (2) (100m backstroke; 50m backstroke)
Ellie Downie (gymnastics)
Giarnni Regini-Moran (all-around gymnastics)
Women's 4x100m medley swim relay team
Abigail Stones (taekwondo)
Men's 4x100m freestyle swim relay team
Sian Rainsley (multi-nation mixed relay triathlon)
Christian McNeish (taekwondo)
Jake Saywell (mixed-nation team equestrian event)
Peter Miles (mixed-nation team judo)
Luke Greenbank (200m backstroke)
Greenbank returned to the water to help Great Britain finish sixth in the mixed 4x100m medley relay alongside Duncan Scott, Charlotte Atkinson and Georgina Evans.
Scott finished sixth in the 100m freestyle final while Evans and Atkinson were eighth in their respective 100m butterfly and 200m breaststroke events.
Zachary Sheridan qualified for the men's trampoline final with the fifth highest score, but would finish sixth after an error ended his routine early.
"It obviously wasn't what I was hoping for but I did achieve a top-eight finish and primarily that was the target," said Sheridan.
Team GB modern pentathlete Francesca Summers made a fantastic start to her campaign by winning 19 of the 23 fencing matches - which placed her third in the rankings.
Team-mate Henry Choong is 14th after winning ten fights and both will return for fencing finals, swimming and the run-shoot combined discipline on Saturday.
All-around gymnastics medallists Giarnni Regini-Moran and Ellie Downie will also be back contesting for individual apparatus honours.