Nuri Sahin sent Brendan Rodgers' men through to round four with an 82nd-minute winner after the Turk had earlier equalised Gabriel Tamas' third-minute opener for the host.
But it was Sinclair - aged 16 years and six days - who stole the headlines when Rodgers opted to send him on moments before Sahin's winner in place of Samed Yesil.
Wayne Rooney returned from injury straight into the Manchester United starting line-up as Sir Alex Ferguson's men saw off Newcastle United with a tight 2-1 win at Old Trafford.
Anderson lashed home a superb 44th-minute opener and when Tom Cleverley added the home side's second in the 58th minute, Ferguson's side seemed to be home and dry.
But Magpies substitute Papiss Cisse came off the bench to reduce the deficit in the 62nd minute and almost grabbed a stunning equaliser when his overhead kick struck the bar.
Arsenal heaped more misery on League One crisis club Coventry City with a rampant 6-1 win at the Emirates Stadium.
The Sky Blues kept the Gunners at bay for 39 minutes until new boy Oliver Giroud netted his first goal for the club to give Arsene Wenger's men a slim half-time lead.
Goals from Alex Oxlade-Chamberlain, Andrei Arshavin, Ignasi Miquel and a double from Theo Walcott sealed Coventry's fate after the break, while Callum Ball's goal gave it some consolation at 4-0.
Tottenham Hotspur avoided any mishaps in Cumbria with a 3-0 win over Carlisle United, the goals coming from Jan Vertonghen in the 37th minute, a stunning Andros Townsend strike on 53, and a late third from Gylfi Sigurdsson.
Reading enjoyed some relief from its tough start in the Premier League as it twice battled back from behind to win 3-2 at Queens Park Rangers.
David Hoilett's early strike for the host was cancelled out by Kaspars Gorkss, only for Djibril Cisse's 71st-minute penalty to put QPR back in front.
But the Royals responded in style with Nicky Shorey equalising five minutes later and Pavel Pogrebnyak - who also missed a late penalty - netting the winner nine minutes from time.
Chris Hughton's youthful Norwich City side edged past League One Doncaster Rovers 1-0 at Carrow Road thanks to a 26th-minute goal from full debutant Alexander Tettey.