The 26-year-old Slovenian-born player saw off his Croatian opponent with a 6-4 6-3 triumph in a match lasting an hour and 15 minutes on Centre Court.
Bedene reached his first ATP final at the Chennai Open in January, where he was beaten by French Open champion Stan Wawrinka, and he admitted he was delighted to advance here.
He wrote on his Twitter account: “Pleased to get the win today here in @CroatiaOpenUmag its one of my favourite tournaments of the year. On to the next round!”
In Monday’s later match, 2013 runner-up Fabio Fognini also made it through in straight sets after seeing off Jiri Vesely.
The 28-year-old Italian – seeded fifth – claimed a 6-4 6-4 victory over the Czech in 92 minutes after saving four of six break points.