At Old Trafford, Manchester United beat Lyon 7-6 on aggregate after a 5-4 extra-time win in their quarter-final second leg. The teams were level at 2-2 after 90 minutes.
Spurs were back in the last-chance saloon and up against it at Deutsche Bank Park after a 1-1 draw in the first leg last Thursday, but produced one of their best displays under Postecoglou to ensure his tenure goes on.
Solanke's spot-kick two minutes before half-time settled the tie and was won after vice-captains Cristian Romero and James Maddison combined.
It was Solanke's first goal in 12 matches and even though Tottenham missed opportunities after the break, they admirably held on to earn a first last-four spot in this competition in 41 years.
Frankfurt, the competition's champions in 2022, enjoyed 61 per cent of possession but Spurs showed plenty of fight.
Postecoglou had admitted to having "no idea" if he would still be in a job after this tie on Wednesday, but can now focus on a two-legged semi-final in May with his team still on course to win the club its first trophy in 17 years.
Elsewhere, Athletic Bilbao advanced after a 2-0 win against Rangers, while Bodø/Glimt lost 3-1 to Lazio but won 3-2 on penalties to earn a semi-final date with Spurs.
Manchester United earlier had blown a two-goal lead before grabbing two goals at the death to advance.
Harry Maguire secured one of the most jaw-dropping European wins in Old Trafford's long history as Manchester United came back to secure a scarcely believable 5-4 extra-time win against 10-man Lyon
There was no room for error as the Red Devils' 14th place in the Premier League standings meant a first European defeat of the campaign would end their hopes of saving a miserable season with silverware and continental qualification.
Corentin Tolisso and Nicolas Tagliafico cancelled out Manuel Ugarte and Diogo Dalot goals to take the match to extra-time on Thursday night, when United were staring down the barrel of a humiliating quarter-final exit.
Lyon made light work of Tolisso's sending off as Rayan Cherki saw his strike followed by Alexandre Lacazette's penalty as the hosts faced the prospect of a first season without European football since 2014-15.
Despite their flaws in recent years, United have frequently shown they can fight until the end.
Bruno Fernandes' 114th minute spot-kick brought hope, substitute Kobbie Mainoo's 120th-minute goal sparked bedlam and moments later Maguire's header sent the home faithful into dreamland, sealing a 7-6 aggregate win and semi-final date with Athletic Bilbao.
In London, Chelsea progressed to the last four of the Conference League but Enzo Maresca's problems continued to mount as Legia Warsaw won 2-1 at Stamford Bridge in their quarter-final second leg to deepen supporters' frustrations with the head coach.
Chelsea won 4-2 on aggregate and are favourites to reach the final in Wroclaw on May 28 but there was little here to strike fear into any opponent.
A penalty by Czech forward Tomas Pekhart gave Legia the lead before Marc Cucurella levelled, but Chelsea were soon undone again when Steve Kapuadi scored early in the second half.
Fiorentina needed Moise Kean's second half goal to draw 2-2 at home against Celje for a 4-3 aggregate win, while Betis advanced 3-1 on aggregate against Jagiellonia Bialystok after a 1-1 draw. Djurgarden beat Rapid Vienna 4-1 away to advance 4-2 on aggregate.
