The British No 1 has become adept not just at coming through the grand slam matches he should win, but doing so efficiently, and so it was here as he defeated the American qualifier 6-4 7-6 (7/0) 6-2.
Edmund did not face a single break point and, although he would have liked a better conversion rate than three of 12 chances on the Klahn serve, there was never a moment when the outcome looked in doubt.
The victory marked another first in a season of them for Edmund as he reached the last 32 at the All England Club for the first time, but he will have his work cut out if he is to go further.
Djokovic has begun to recover some of his old swagger and has cruised into the third round, although Edmund did win their most recent match on clay in Madrid in May.
Their next meeting will come tomorrow.
Edmund walked onto Centre Court shortly after British women’s No 1 Johanna Konta had fallen to defeat, which made him the last home player standing in singles for the fourth successive slam.
The Yorkshireman’s task looked significantly less formidable than Konta’s, with Klahn ranked at No 168 and through to the second round of a slam for the first time since 2013 after two years lost to back surgery.
Klahn had been hoping for a Centre Court date, but early nerves were evident as he double-faulted three times in a nine-minute opening game and Edmund broke.
The American battled hard to stay in contention in the set without ever threatening to break, and it was a similar story in the second.
He must have hoped by pushing it to a tie-break he could find a chink in the Edmund armour, but far from it, the home favourite showing an impressive ruthlessness by taking it to love.
Edmund finally got his second break of the match in the sixth
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