The British number two came out on top 6-3 6-4 7-6(5) in two hours and 34 minutes to put his nation into an unassailable 3-1 overall lead in Belgrade.
Edmund, fresh off the back of winning Friday's opening singles match, held throughout the first set and was able to break his opponent in the sixth game to race ahead.
Having dropped just seven games on his way to beating Janko Tipsarevic earlier in the week, Edmund again held his serve throughout the second set and once more broke his opponent to put Britain into a commanding lead in the match.
Lajovic did prolong his hopes of a comeback when eventually breaking his opponent in the 10th game of the final set, but Edmund was in no mood to hold back and managed to get over the line with his third set point.
Victory for Jamie Murray and Dom Inglot in Saturday's doubles match, combined with Edmund's double singles success, ensures that Britain will now face Argentina at home in September as their defence of the cup goes on.