Andy Murray was left out of the GB singles line-up following his sluggish victory against 179th-ranked Dutchmen Tallon Griekspoor as GB beat the Netherlands, and Edmund rose to the challenge overcoming Mikhail Kukushkin in one hour and 16 minutes.
The British number two converted a fourth break point in the fifth game of the first set before serving it out to take a one-set advantage, eliciting big applause from Murray on the side of the court.
Britain's Draper collapses on ATP Tour debut
Leon Smith, Captain of Great Britain applauds Kyle Edmund
Image credit: Eurosport
Edmund fended off two break points early in the second courtesy of an ace and winner to a hold in front of an enthusiastic crowd, roaring at coach Leon Smith with delight.
And the Brit punished his opponent almost immediately as Kukushkin served up two break points, Edmund taking the second after the Kazakh swatted a forehand into the net.
Edmund then sailed towards a win, holding serve with ease before earning three match points on Kukushkin's serve, snatching the first and giving GB the edge in the tie and one foot in the quarter-finals.
British number one Dan Evans can put Britain into the quarter-finals if he beats Alexander Bublik later today, followed by a potentially decisive doubles rubber should Evans lose.
'Very, very special' - Thiem wins spectacular point against Kukushkin
Thiem into round two after beating plucky Kukushkin