Edmund carried Britain into the semis with a 6-3, 6-4, 7-6 (5) win over Lajovic.
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Leon Smith’s squad will now face a home tie against Argentina, who took an unassailable 3-1 lead against Italy in Pesaro.
Marin Cilic set up the deciding fifth match by beating John Isner 7-6 (9), 6-3, 6-4 in the first reverse singles match on Sunday after combining with Ivan Dodig on Saturday for a win over twins Bob and Mike Bryan in doubles to keep Croatia alive in the contest after the Americans swept the opening singles matches.
The 26-year-old Serbian was also affected by weather conditions that were unusual for this time of the year for Belgrade – cloudy, foggy and cold.
Lajovic played his part in the contest but, just as it appeared he was about to claim the third set, world No 67 Edmund hit back to secure one of the most important wins of his career.
In the last and meaningless match, Tipsarevic beat James Ward 6-2, 3-6, 7-5.
Kyle Edmund won his second singles rubber of the weekend to steer Davis Cup champions Great Britain past Serbia on Sunday and into a mouthwatering semi-final clash against Argentina.
Andy Murray may well return for the semi-final, making Britain firm favourites to reach the final once again after their historic victory previous year.
And a disappointed Lajovic was magnanimous in defeat, adding: “It’s a big difference if Andy plays or not”.
It was throwing caution to the wind to play two of the European ties on clay this summer: Torrential rain briefly hit the Rome Masters, nearly washed out the first week of the French Open, wrecked havoc on the grass of Halle, Stuttgart and Queen’s, and rearranged the schedule of the first three rounds at Wimbledon.
Asked whether it would be on his schedule he said in a convincing tone: ‘I think so, yeah’.
However, Edmund had to survive a third set wobble when he served for the match at 5-4, only to be broken for the first time.
“He’s improved so much physically”.
“It’s great to showcase our sport again, especially later in the year after our grass season has finished and we’ve got this huge event now. His serve is better, his backhand is rock solid”.
Edmund had only faced one break point in the whole match but his first serve deserted him and he tightened up on his forehand as Lajovic levelled. “The forehand is his best weapon and it made the difference today”. There are a lot of times you come off court and you’re a bit down on yourself because you feel the other guy played better.
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There is no rest for Edmund, who heads to Toronto on Tuesday to play in qualifying for the Rogers Cup – which Murray looks set to miss – before making his Olympic debut in Rio.
