Bodo/Glimt had won the first leg 2-0 in Norway
Bodo/Glimt survived a superb Lazio fightback to win a frantic encounter on penalties and reach the Europa League semi-finals.
The 3-2 success on spot-kicks, following a 3-3 draw on aggregate, was the first time any Norwegian side had reached the last four of a men's European competition.
Bodo/Glimt will now face Tottenham for a place in the final, with the first leg taking place in London on 1 May and the second at the Aspmyra Stadion on 8 May.
With a two-goal deficit to overcome, Valentin Castellanos gave Lazio hope in the Rome return with a 23rd-minute opener.
The Argentine forward sent an expert backheeled effort past Nikita Haikin after Gustav Isaksen pulled the ball back to him on the edge of the six-yard box.
Tijjani Noslin then made it 2-0, levelling the tie on aggregate in dramatic fashion, when he bundled in from close range from a corner in the third minute of injury time.
Substitute Boulaye Dia added a third in extra time to put Lazio in control of the tie, leaping highest to head home a Matteo Guendouzi cross.
However, Andreas Helmersen brought the visitors back to parity in the 109th minute by powering Sondre Brunstad Fet's cross past home keeper Christos Mandas.
The forward, though, could not take part in the resulting shootout when he was sent off for a second yellow card in the final minute of extra time.
Jens Petter Hauge had Bodo/Glimt's first penalty saved while Lazio scored through Dia, but two missed spot-kicks from Loum Tchaouna and Noslin, and two scored by Fet and Sondre Sorli swung the outcome in favour of the Norwegian side.
But Patrick Berg missed the chance to win it for the visitors after both sides had netted their fourth penalty.
But goalkeeper Haikin then saved Castellanos' effort to make it 3-2 to Bodo/Glimt and spark jubilant celebrations for their historic achievement.
