Liverpool, whose win came thanks to an early Martin Skrtel goal, a neat Andy Carroll finish, three own goals and a late Luis Suarez header, next host last term's runners-up Stoke City in the quarter-finals after they prevailed 2-0 at Crawley Town.
Spurs, third in the Premier League and with coach Harry Redknapp tipped to take over England, produced a lethargic display with three centre-backs and had to settle for a replay at White Hart Lane next month.
"At the end of the game it got stretched and we were on the front foot. I think we did our homework, they had possession but we had a couple of chances," Day told ITV.
"[The replay] will be a long shot, it will be first time I've been down there this year without having to pay to get in," he joked.
London's three big clubs have struggled in the fifth round with Arsenal being eliminated by Sunderland - who next visit Everton - and Chelsea scraping a 1-1 home draw with Birmingham City on Saturday with Leicester City awaiting in the last eight.
Liverpool, facing Cardiff City in next Sunday's League Cup final, took a fifth-minute lead when Skrtel headed in Steven Gerrard's corner but a Kazenga LuaLua cracker from a free-kick brought second-tier Brighton level.
A comical Lewis Dunk own goal and a close-range header from Suarez, who earlier missed a penalty having been included despite his non-handshake with Patrice Evra last week, completed the rout against his Uruguayan compatriot Gus Poyet's side.
