Trump will face Michael Holt in the third round after making breaks of 50, 132 and 54 against the outclassed German world number 108 at the Marshall Arena in Milton Keynes.
While Trump took only 16.6 seconds per shot, Kleckers averaged 29.9 seconds in failing to register any breaks over 50 in the match.
“It was important to get off to a good lead. He is quite a slow player," said Trump, who lifted a record six ranking events last season.
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"It can be frustrating at times if you are sat there watching. I just wanted get it done as quickly as possible.
“You have to win one of the first couple of events to get that confidence up. When you start a new season it is so important to have a quick start and get a few wins under your belt. I feel like my game is there and I feel like I’m playing well.
"Hopefully I can show that for the rest of the tournament.”
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