City's leading ALM scorer increased his tally for the season to eight when he broke the deadlock in the 53rd minute.
Caputo jumped over the nine centimetres smaller Grant to plant a firm header into the net from a left wing corner from Daniel Arzani.
It was the decisive moment in Grant's 400th appearance for Sydney and City's 500th competition match.
City's first win in six ALM games moved them from 10th to seventh, just two points behind sixth-placed Central Coast.
Their unbeaten ALM run against Sydney increased to nine matches, with the Sky Blues' last victory against them in December 2022.
It was Sydney's first loss in five ALM games, but City were good value for their victory.
They had three more attempts on target and created the best chances.
Austrian midfielder Andreas Kuen rattled the crossbar with a fierce shot from just outside the penalty area and Congolese defender Marcel Tisserand cleared off the line.
The ball appeared over the line from at least one camera angle, but VAR decided it was not a goal.
"It was a big game for us, probably a season-defining game for us," City coach Aurelio Vidmar told Paramount+.
"I thought we thoroughly deserved the three points tonight."
Grant had a great battle out wide with fellow Socceroo Arzani and the Sydney man had more chances to score than most of his colleagues.
He had two attempts saved by goalkeeper Patrick Beach and put a header wide.
City looked the more likely to score after going ahead, with the lively Marcus Younis fizzing a shot just wide and Elbasan Rashnai having an attempt deflected away by Sydney goalkeeper Harrison Devenish-Meares.
Sydney remain third, six points behind second and only a point above the two teams just below them.
"We had some good transition moments to go and capitalise on, especially the first half (but) we didn't," Sydney coach Ufuk Talay told Paramount Plus.
"The second half, I thought we were a bit lacklustre and lacked intensity to get something out of the game."
Talay was adamant speculation linking him to the Western Sydney Wanderers coaching position for next season was not having an adverse affect.
"I don't think it's a distraction for me or the team whatsoever," Talay said. "We need to focus on what we want to achieve."
