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The 33-year-old improved in the second set but again dropped serve early on and was chasing from then on.
With just under an hour and 10 minutes on the clock, Ofner was 30-0 up serving for the match but offered Evans a glimmer of hope with a couple of errors to bring it back to 30-30.
However, the 27-year-old Austrian regained his composure to seal his place in the second round, 6-1 6-4.
Evans lost his first clay-court match of the season in straight sets to Roberto Carballes Baena at the Grand Prix Hassan in Marrakech last week.
Dan Evans suffers another straight sets defeat on clay