Loughborough ITC swimmer Burnell finished eighth in 1:50:49 hours, just 11 seconds behind third-place German Thomas Lurz who has taken 10km open water silver and bronze from the last two Olympics.
Frenchman Romain Beraud claimed the victory in 1:50:21 hours, nine seconds ahead of compatriot Damien Cattin Vidal.
But there was more for Great Britain to shout about as they had two males finish in the top ten with Tom Allen finishing six seconds behind Burnell in tenth, with Tom Sunter just one place further back after crossing the line in 1:51:57 hours.
In the women's race Rachel Williamson finished 15th in 2:30:19 hours while compatriot Lucy Charles was forced to withdraw through shoulder injury.
Berni Dietzig, Interim Head of Open Water, said: "As we expected, this trip wasn’t easy. The journey from the first leg in Santos to Viedma took 18 hours.
"With only two days on the ground before race day the team followed a stringent preparation program consisting of course familiarisation, training, mobility and stretching routines in order to get race ready in time.
"The Viedma 10 km was swum in a river which is tidal and has extremely strong currents so our swimmers were challenged to find the best lines.
"Having learned valuable lessons in Santos, Burnell, Allen and Sunter swam a tactically aware race, dealing with breakaways and tidal currents and expertly managing to get themselves into the lead group for the final sprint.
"Considering the international depth of the field, we are very happy with an 8th, 10th and 11th finish for Brunell, Allen and Sunter respectively.”