Held 1-1 at home in the first leg of their last 16 tie, the Turkish league leaders fell behind, hit back to lead 2-1 at half-time, spent the entire second half with their backs to the wall and then snatched a 3-2 win in stoppage time.
Their brilliant win put them in the Champions League quarter-finals for the first time since 2001 and began to pay back the huge investment made in Dutch playmaker Wesley Sneijder and Ivory Coast forward Didier Drogba.
Terim, who has also coached Fiorentina and AC Milan, is in his third stint as Galatasaray coach but Tuesday's display was more reminiscent of his days with the Turkish national side, in particular the Euro 2008.
Having lost their opening group game, Turkey came from behind to beat co-hosts Switzerland 2-1 and then hit back from two goals down, scoring three times in the last quarter of an hour, to beat the Czech Republic 3-2.
Even more dramatic was the quarter-final win over Croatia, where Turkey fell behind in the 29th minute of extra-time, equalised immediately and won on penalties.
Galatasaray went into Tuesday's match on the back of a shock 1-0 defeat to Genclerbirligi, when Drogba missed a penalty and Burak Yilmaz, who scored their second on Tuesday, contrived to miss an open goal from six metres.
The contrast on Tuesday could not have been more different as Terim fielded an attacking line-up with Drogba and Yilmaz up front and Sneijder playing just behind them.
With eight goals, Yilmaz is the competition's joint leading score alongside Real Madrid's Cristiano Ronaldo.
"We played like an excellent European team in the first half," said Terim.
"Players sometimes known for their limited movement and tackling ability were transformed in this game and fought for the team on the pitch."
Despite not scoring, Drogba, with his unmistakable physical presence, was inspirational although Sneijder is still struggling to find his best form and will need to improve if Galatasaray are to progress to the semi-finals.
In midfield, Felipe Melo, the villain of Brazil's 2010 World Cup campaign when he was sent off in the quarter-final defeat to Sneijder's Netherlands, won more tackles than any other player on the field.
Channelling his aggression more positively than earlier in his career, he committed only three fouls and avoided a yellow card.
Drogba's deal included transfer fees totalling six million euros and payments of 15,000 euros per match, plus signing fees of four million euros.
Drogba, who scored the equaliser for Chelsea in their 1-1 draw with Bayern Munich in last year's final before netting the decisive penalty in the shootout, moved to Shanghai Shenhua but cut short his stay in China amid reports of unpaid wages.
Sneijder signed a three-and-a-half-year contract with Galatasaray and will cost the club around four million euros a year.
Galatasaray returned to predictably jubilant scenes at Istanbul airport and Turkish Prime Minister Tayyip Erdogan called Terim to congratulate him on the victory, state-run Anatol
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