By the standards of their last two League games, both won in stoppage time, the Premiership leaders scored early. Fully five-and-a-half minutes remained when Radebe rose to meet Stephen McPhail's far-post corner to make Harry Kewell's goal in the "away" leg in Sofia count double for Leeds.
Radebe, who has now scored three times in his Leeds career, is making a habit of striking in this season's Uefa Cup. The South African defender also found the net against Partizan Belgrade with an overhead kick while lying on his backside.
By inflicting Spartak's first-ever defeat by an English club - and adding their scalp to that of Lokomotiv Moscow - David O'Leary's young team have now advanced further in Continental competition than any Leeds side since Billy Bremner and company reached the European Cup final in 1975.
Moreover, it was the first time since the 1966-67 season that Leeds, under Don Revie, had won a European tie after losing the first leg. For O'Leary, the satisfaction of prolonging their participation was compounded by the fact they had knocked another club drafted in - unfairly in his view, like Partizan - from the Champions' League.
Leeds' victory also avenged what O'Leary described as "the biggest chasing of my playing career", when a Spartak side containing the current coach, Oleg Romantsev, embarrassed Arsenal 5-2 at Highbury in 1981.
Romantsev refused to attend the post-match press conference in protest against Leeds' alleged involvement in having the original first-leg match in Moscow postponed. His assistant, Viacheslav Grozny, appeared instead to accuse Leeds of foul play. "We feel that if we'd had a normal home leg, we would have come out on top," he complained.
The Russians would have had a stronger case had they used the undersoil heating in the Dynamo Stadium. Temperatures in Moscow were -22C and the referee had no hesitation in ruling the surface unplayable.
Elland Road was in pristine condition for the return, suiting Spartak's superior touch play. Leeds would have fallen further behind in the fifth minute had Nigel Martyn not reacted quickly to block with his feet a shot by Luis Robson, the Brazilian striker, from inside the six-yard area.
Leeds retaliated strongly, without being at their most fluid, and Michael Bridges diverted Ian Harte's inswinging free-kick against a post midway through the first half. Spartak had to withdraw their goalkeeper, Alexander Filimonov, after he fell heavily shortly before half-time, but Andrei Smetanin proved an defiant and agile deputy.
Smetanin's best save, from a Bridges free-kick after 70 minutes, seemed likely to condemn Leeds to a frustrating exit. But Spartak had reckoned without Radebe and, as O'Leary purred: "We just kept going... and going."
Leeds United (4-4-2): Martyn; Kelly, Woodgate, Radebe, Harte; Bowyer, McPhail, Bakke, Kewell; Smith (Huckerby, 72), Bridges. Substitutes not used: Haaland, Mills, Jones, Duberry, Robinson (gk).
Spartak Moscow (1-2-5-2): Filimonov (Smetanin, 38); Bushmanov; Khlestov, Evseyev; Baranov, Parfionov, Titov, Bulatov, Tikhonov; Robson, Schirko. Substitutes not used: Ananko, Ketchinov, Penichev, Meleschin, Mor, Bezrodniy.
Referee: A-J Lopez Nieto (Spain).Reuse content