Rodgers proud of Celtic for 'playing like men' after Old Firm win
Kyogo Furuhashi's goal - the Japan star's sixth in five starts in Glasgow's Old Firm derby - in first-half stoppage time proved the difference at Ibrox.
Celtic, who had rejected the offer of 700 tickets for away fan on safety grounds, were forced on the defensive late in the game to preserve their lead.
"It's a massive result for the squad when you consider virtually half the team is unavailable," Rodgers told the BBC.
"I said to them before the game that I need my men today. You need to come here, with 50,000 baying for blood, you need to come and play like a man, even though you're really young players."
'Resilience'
Rodgers, in his second spell as Celtic manager, added: "Second half, we had to show resilience and fight. To come here, keep a clean sheet, and win with not one supporter in the stadium is a brilliant win."
Rangers almost went ahead in the first minute but Rabbi Matondo's effort was ruled out for offside.
More controversial was the decision to disallow a goal by Rangers' Kemar Roofe following Cyriel Dessers's foul on Gustaf Lagerbielke.
Rangers, fresh from their midweek 5-1 hammering by PSV Eindhoven in Champions League qualifying, desperately needed a win in front of their own fans at Ibrox.
Instead, they find themselves already four points behind leaders and reigning champions Celtic ahead of the international break, with boos ringing round the ground at the final whistle.
By contrast, victory eased some of the mounting pressure on Rodgers following a Scottish League Cup defeat by Kilmarnock and a home draw with St Johnstone.
Following Matondo's disallowed effort, Celtic almost went ahead in the 16th minute but Furuhashi was unable to make a clean connection with Liel Abada's cross.
Rangers appeared to have gone ahead after Celtic's Lagerbielke was challenged by Dessers just inside the Hoops half and the ball broke clear for the Rangers forward.
He then squared the ball for Roofe, who beat Celtic goalkeeper Joe Hart.
But when referee Don Robertson checked the pitchside monitor, he ruled the goal out for a foul on the Celtic defender.
"I think it's a harsh call," said Rangers manager Michael Beale. "But there's a lot of time to play in the game."
Celtic broke the deadlock seconds before half-time when midfielder Matt O'Riley returned a header from Rangers' Connor Goldson into the path of Furuhashi, with the Japan striker driving the ball past home goalkeeper Jack Butland.
Celtic's failure to score a second goal almost proved costly, with Hart racing out to deny Sam Lammers before twice saving from close range to frustrate Danilo.
"Second half, we pushed and pushed, but you'd like us to be more inventive and ruthless because the chances were there," said Beale. "It's a desperately disappointing day for everybody. It's been a rough week."