Stand-in City skipper Nabbout scored the winner in the 49th minute but went down with a seemingly serious right knee injury five minutes later at AAMI Park.
It left coach Aurelio Vidmar counting the cost of City's otherwise impressive response to their derby loss to Melbourne Victory.
City were already without players including Max Caputo, Arion Sulemani and Alessandro Lopane through injury - but welcomed back Mathew Leckie in the 78th minute.
On-loan Celtic attacker Tilio, was distraught when forced from the field after going down in the 38th minute.
Tilio has had a rotten run over the past two years, suffering multiple hamstring injuries last season after previous quad issues.
Socceroos coach Tony Popovic was among those watching when Tilio broke away through the middle of the park, was knocked off balance in a challenge and immediately grabbed his left hamstring when he hit the ground.
Tilio sank his face into his hands as he limped off the pitch and down the tunnel, replaced by debutant Medin Memeti.
With captain Aziz Behich suspended, City coach Aurelio Vidmar turned to Nathaniel Atkinson at left-back, with the fringe Socceroo impressing throughout.
City dominated the first half, but United's Rhys Bozinovski had a shot deflected over the crossbar in the 31st minute while Riku Danzaki flashed a late shot into the side netting.
City took the lead when Nabbout forced a turnover then burst forward.
Memeti slipped through Andreas Kuen, who squared it for Nabbout to tap home.
But five minutes later, Nabbout hurt his right knee when landing as he attempted to win the ball against United defender Charbel Shamoon.
The explosive attacker has also had bad luck with injuries, most recently suffering an achilles problem in 2023.
United brought on Abel Walatee and Noah Botic just after the hour mark and immediately started looking dangerous, with Bozinovski and Danzaki going close to scoring.
Western wanted a penalty when Atkinson appeared to handball in the area in the 67th minute but their pleas were waved away.
For all their efforts, United couldn't manufacture an equaliser, with Hiroshi Ibusuki flicking a late header over the bar.
Fourth-placed City (2-0-1) will regain midfielder Jimmy Jeggo from suspension for next Sunday's away game against Perth Glory while 10th-placed United (0-2-1) play Adelaide away on Saturday.
