Rush swooped for his 23rd goal of the season in the second period of extra time to set up a semi-final tie with Southampton. Everton play Sheffield Wednesday in the other semi.
But Liverpool can thank their lucky stars, because they equalised only four minutes from the end of normal time when Jan Molby scored from the penalty spot after 'keeper Tony Coton had brought down Rush.
Liverpool's early football showed enough promise with Dalglish giving Molby the opportunity of a 25 yard shot which again Coton handled with impressive concentration.
Watford, with skipper Brian Talbot showing an aggressive appetite at last, launched their first serious assault when Colin West headed down Kenny Jackett's free-kick for Lee Sinnott to test Bruce Grobbelaar with a first time shot.
Then the home fans screamed for a penalty in the 26th minute when West felt he had been pushed in the back by Gary Gillespie, but Bristol referee Roger Milford decided the Watford striker had exaggerated his fall.
The second half could hardly have begun more explosively. Within a minute the referee had his notebook out to caution Beglin as he clearly held back Sterling on the edge of the Liverpool area.
But the greater punishment came for Liverpool from the resultant free-kick as John Barnes curled the ball viciously round the Liverpool wall and past Grobbelaar.
In the exciting few minutes that followed Molby also went into the notebook for dissent and within a minute Dalglish was aiming a shot just wide of Watford's far post.
Barnes almost repeated the trick in the 56th minute, this time from 30 yards, though. Grobbelaar managed, this time, to get to his post in time to collect. But then as Watford fans were preparing to celebrate a semi-final place, in stepped Molby to take the game into extra-time.
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