The eighth seed has been plagued by a foot injury since the Australian Open in January, and was also looking to exorcise some SW19 demons having injured her knee on court one 12 months ago.
Azarenka was forced to save set points in the second set, however Lucic-Baroni could not convert as the two-time major winner ran out a 6-3 7-5 win in 94 minutes.
Chinese second seed Li Na was a convincing winner on Monday as she put an early wobble against Poland's Paula Kania behind her to come out a 7-5 6-2 winner.
Straight-set wins were the order of the day for the seeds as Petra Kvitova (six), Flavia Pennetta (12), Lucie Safarova (23) and Elena Vesnina (32) also progressed without dropping a set.
Five-time champion Venus Williams picked up her first win at Wimbledon since 2011 as she battled to a 6-4 4-6 6-2 victory over Spain's Maria-Teresa Torro-Flor, while Russian Ekaterina Makarova edged out Japanese veteran Kimiko Date-Krumm 3-6 6-4 7-5.
Things were not so positive for seeds Samantha Stosur and Sloane Stephens, though, as they were the shock losers on day one.
Stosur, the 17th seed, was comfortably beaten 6-3 6-4 by Belgium's Yanina Wickmayer, and Russia's Maria Kirilenko had too much for 18th seed Stephens 6-2 7-6 (8-6).
Naomi Broady is the only Briton left in the draw after she came from behind to beat Hungary's Timea Babos 2-6 7-6 (9-7) 6-0, but her compatriot Johanna Konta fell to a 6-4 3-6 6-4 loss to China's Peng Shuai.
American Coco Vandeweghe continued the form that saw her win the Topshelf Open ahead of the grand slam as she toppled Spanish 27th seed Garbine Muguruza 6-3 3-6 7-5, while there were also victories for Australia's Casey Dellacqua and Croatia's Ana Konjuh.
Serbia's Bojana Jovanovski is Azarenka's next opponent after she accounted for Argentina's Paula Ormaechea 7-6 (7-2) 6-0, while Austria's Yvonne Meusberger and German Mona Barthel will meet Li and Kvitova respectively.
There were straight set wins for Slovenia's Polona Hercog, American Lauren Davis, Czech Barbora Zahlavova Strycova and Japan's Misaki Doi.
A clash with Venus Williams is next up for Japan's Kurumi Nara after her win over Germany's Anna-Lena Friedsam, with Czech Tereza Smitkova and Australia's Jarmila Gajdosova also progressing, and Belgium's Alison Van Uytvanck will play Slovakian 10th seed Dominika Cibulkova following their straight set wins over Monica Niculescu and Aleksandra Wozniak respectively.
Rain late on Monday saw five games suspended with Polish fourth seed Agnieszka Radwanska 4-2 up on Romania's Andreea Mitu.
Denmark's 16th seed Caroline Wozniacki was 6-3 2-0 up against Israel's Shahar Peer when the rain came, Italian 14th seed Sara Errani was level at one-set all with France's Caroline Garcia and Russia's Svetlana Kuznetsova and Portugal's Michelle Larcher De Brito had seen their first three games go with serve.
Bulgaria's Tsvetana Pironkova is leading 7-6 (8-6) 0-2 against American Varvara Lepchenko in another match to be held over to Tuesday.