Bopanna and Myneni prevailed 7-6 (4), 5-7, 7-6 (2), 6-3 to give India a 2-1 lead after day one had ended 1-1.
"Great effort and win by @rohanbopanna and saketh to get the doubles point and give us a 2 1 lead. Way to go boys. #DavisCup #TeamIndia," tweeted Sanam Singh, who had lost in the opening singles yesterday, after the match.
India are now on the verge of beating Korea for the first time in an away tie. They also have a chance to improve on the overall head-to-head record which stands at 3-6.
The last time the two teams played was in 2013 when the Koreans beat India 4-1 in New Delhi in the absence of top players, who boycotted the tie.
India are coming into this tie after having blanked Taipei 5-0 while the hosts had got a bye in the first round as one of the top seeds and are seeking a World Group play-off berth for the first time since 2008.