Higuain was left out of Milan's matchday squad for their first Serie A outing of 2019 due to concerns from head coach Gennaro Gattuso – who watched from the stands due to a suspension – over his mindset amid links with a move to Chelsea.
Gattuso, punished for angrily confronting the referee after his side's Supercoppa Italiana loss to Juventus last week, saw the Rossoneri toil for much of the game.
Milan had previously fired blanks in successive away league matches, but Borini broke the deadlock in the 72nd minute and Suso guaranteed a return to fourth place in the closing stages.
Already without Alessio Romagnoli and Mattia Caldara at the back, Gattuso was dealt an early blow when Cristian Zapata went off injured and was replaced by Andrea Conti in the 14th minute.
Goran Pandev - selected in place of supsended Milan target Krzysztof Piatek - and Daniel Bessa threatened, the latter seeing a shot kept out by Gianluigi Donnarumma when through on goal as Genoa took control of the contest.
Lucas Paqueta nearly fired the visitors in front late in the first half with a well-struck volley that skimmed the outside of the post, shortly after Domenico Criscito's deflected cutback was blocked by Donnarumma.
Borini saw an angled drive tipped away by Ionut Radu, while Lucas Paqueta failed to make the most of a free header before Donnarumma's double save denied Christian Kouame and Bessa.
Milan did find a way through eventually, however, as an unmarked Borini got on the end of Conti's cutback and converted from eight yards.
The Rossoneri survived a scare when Donnarumma touched Miguel Veloso's shot onto the crossbar, but Suso wrapped up the points in the 83rd minute by latching onto Cutrone's throughball and finishing off the inside of the post.
Higuain absence no issue
An improved second-half showing from Milan and an end to their three-match winless run on the road in Serie A takes them back into the top four.
All eyes were on Gattuso's attacking players and Borini and Suso stepped up thanks to their strikes in the final quarter of the match, making it back-to-back league wins for the first time since the start of November.
Donnarumma produces heroics
A victory for Milan might not have been possible without the heroics of Donnarumma. The 19-year-old goalkeeper produced a total of seven saves, including a fingertip stop to help Veloso's 25-yarder onto the frame of the goal just nine minutes before Suso sealed the win.
Cutrone the odd man out
Cutrone retained his place from the side that lost 1-0 to Juventus in the Supercoppa Italiana last week but was noticeably the only member of Milan's attacking trio not to register a goal against Genoa, attempting just 10 passes all match. However, one of those did lead to his side's killer second goal.
