Palmer opened the scoring just four minutes after Seagulls’ goalkeeper, Fynn Talley, had denied him from 12 yards and showed nerves of steel to take responsibility for and convert our second spot kick as City made a flying start to the second half.
Robertson netted the match winner 16 minutes from time after Brighton's Lorant Tolag had pulled one back on the hour mark and though he scored again in the 90th minute to set up a nerve-jangling period of added time, City held on to progress.
How we lined up
What it means
City are one game away from making the Premier League Cup final for the second consecutive year and could face Chelsea, Stoke City or Reading in the semi-finals.
League action returns next weekend, when we will look to extend our eight-match unbeaten run at home to Middlesbrough. Kick-off is at 12:30 (UK).