Southampton were upstaged 2-0 by Championship outfit Wolves, who progressed thanks to Danny Batth's 67th-minute header and a late sealer from substitute Donovan Wilson at St Mary's.
Forest needed extra time to advance 3-2 against Rafael Benitez's Newcastle, Tyler Walker – son of former England defender Des – scoring the winner past ex-Forest goalkeeper Karl Darlow seven minutes into the added period at St James' Park.
Jason Cummings had earlier scored a quick-fire brace for Forest in an eventful first-half, turning the game around for the visitors after Aleksandar Mitrovic's early opener, with Achraf Lazaar levelling the scores again approaching the break.
In the East Lancashire derby, Robbie Brady's cross was glanced in by Burnley team-mate Jack Cork for the opener and the two players combined again shortly before half-time, Cork feeding Brady, who neatly evaded his marker to crash home a left-footed drive.
While Burnley extended their unbeaten run against Blackburn to seven matches, there could be repercussions for the League One club after two Blackburn supporters ran on to the Ewood Park pitch to confront visiting players after Cork's goal.
West Ham United brushed off a pointless start to the Premier League season with a 2-0 win at Cheltenham Town thanks to goals from Diafra Sakho and Andre Ayew in a five-minute spell before the interval.
Stoke City also cruised into the third round via a 4-0 battering of Rochdale, Joe Allen with a first-half brace for the Potters, while goals from Philip Billing and Joe Lolley saw Huddersfield Town come from behind to beat Rotherham United 2-1.
Clubs will find out their third-round opponents when the draw is made in China on Thursday, with teams involved in European competitions entering the competition at this stage.