The 15th seed dropped serve early on the way to losing the opening set but raised his game impressively after that to win 3-6, 6-3, 6-2, 6-2.
Rune, who reached the quarter-finals last year, will face French qualifier Quentin Halys next after he knocked out 21st seed Karen Khachanov in five sets.
Regarded as a player likely to one day challenge for a Grand Slam title, Rune did not let the unpredictable Seyboth Wild or a gusty wind on Court 18 throw him off balance.
He delighted fans, many perched on the sunny terrace overlooking one of Wimbledon's most quirky courts, with his full repertoire of powerful groundstrokes and silky touch, drawing gasps with one incredible running backhand pass late on.
Seyboth Wild also contributed to the spectacle with some dazzling strokes of his own but was eventually outclassed.
