Defending champion Aston Villa was also eliminated, going down 4-1 at Crystal Palace on a pair of goals from both Chris Armstrong and Gareth Southgate.
Three Division One teams advanced to the quarterfinals, two with upsets of Premier League clubs. Greg Berry scored both goals as Millwall won 2-0 at Nottingham Forest, and John McGinlay netted twice in Bolton's 3-1 victory at West Ham.
Former England midfielder Steve McMahon made a successful managerial debut as Swindon beat fellow Division One club Derby 2-1, while Arsenal and Norwich also joined the final eight. One game, Newcastle versus Manchester City, will be replayed in three weeks after the teams drew 1-1.
At Blackburn, Liverpool captain Rush scored his 44th, 45th and 46th career League Cup goals in the 19th, 57th and 73rd minutes. The Welsh striker needs three more goals to match Geoff Hurst's all-time record of 49.
Rush's first was a screaming left-footed drive from the edge of the box off Robbie Fowler's pass. Rush took Rob Jones' pass and beat defender Henning Berg to score the second, and the third came after Graeme Le Saux mis-headed a corner kick clearance, leaving the Welshman to slot the ball between the legs of goalkeeper Tim Flowers.
Blackburn, which had been riding a six-game winning streak, got a consolation goal when Chris Sutton scored his 17th of the season, a header off Stuart Ripley's cross. The Birmingham club still managed to take the lead on Dalian Atkinson's goal in the 32nd, but Armstrong netted in the 48th and 88th, and assisted on Southgate's score in the 58th. Southgate added another in the 75th.
Irish midfielder Steve Morrow scored in the 29th and Ian Wright got his league-leading 18th goal in the 34th as Arsenal overcame a week of off-the-field controversy to defeat Sheffield Wednesday 2-0 at Highbury.
Striker Paul Merson's admission of cocaine use has rocked the London club, and league officials are also looking into alleged financial irregularities involving the team's acquisitions of foreign players. In addition, Arsenal was winless in its last three games.
At West Ham, McGinlay opened the scoring for Bolton in the 16th and an Adrian Whitbread goal made it 2-0 in the 53rd. McGinlay's second was a penalty in the 76th after Julian Dicks was whistled for a hand ball. West Ham's goal came from Tony Cottee in the 81st.
At Swindon, Jan-Aage Fjortoft scored his 16th and 17th goals of the season in the eighth and 84th minutes and in the victory over Derby, which netted through Mark Stallard in the 35th.
At Norwich, Darren Eadie's free-kick goal, his first score of the season, in the first minute was the winner in a 1-0 win over Notts County, the last-place team in Division One.
Berry's goals came in the 18th and 41st in Millwall's victory at Nottingham Forest. Forest, which hasn't scored in six consecutive games, lost defender Colin Cooper and Norwegian international Lars Bohinen to injuries.
In the other fourth-round game, goals by Newcastle's Mike Jeffrey, in the 11th, and Manchester City's German substitute Uwe Rosler, in the 69th, canceled each other out at City's Maine Road.
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