Former world number six Simon let 11 break points slip and Cecchinato converted six of them to secure a 6-1 3-6 6-1 win for his first main-draw victory on the ATP Tour since April.
The Italian will take on Croatian wildcard Ivan Dodig in the second round after he saw off Marius Copil 6-4 7-6 (7-4) on Monday, with a quarter-final clash against top seed David Goffin likely to await the winner.
In Newport, a brilliant third set from 178th ranked Kudla rounded off a 6-4 5-7 6-0 victory over Mitchell Krueger and sent him into a second-round match with second seed and 2016 winner Karlovic.
Taylor Fritz was not as fortunate as fellow American Kudla with a 6-3 6-3 loss to Tobias Kamke sending the German through to face third seed Adrian Mannarino, while Adrian Menendez-Maceiras overcame Stefan Kozlov 6-2 7-6 (7-5) for a match-up with Pierre-Hugues Herbert.
It was not good news for eighth seed Illya Marchenko, who was upstaged 7-6 (7-2) 1-6 6-4 by Bjorn Fratangelo.
In the only main-draw match at the Swedish Open, home hope Mikael Ymer twice failed to serve out the opening set as he fell to a 7-6 (7-5) 7-5 defeat to Henri Laaksonen. Next up for the Swiss will be third seed Pablo Cuevas.
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