Spurs were also up against it heading into their second leg, having lost 1-0 at home to Ajax in their opening leg.
And, at halftime, the task was even bigger as Ajax built a 2-0 lead on the night on goals from Matthijs de Ligt and Hakim Ziyech.
But Spurs exploded in the second half with finishes from Lucas Moura and in the 55th and 59th minute pushing Tottenham right back into the tie.
And, in the 96th minute, Moura popped up again, scoring Tottenham's third and final goal to level the score at three apiece and send Tottenham to the finale on way goals.
Rose says their final opponents, Liverpool, proved an inspiration, one that showed that that comebacks like Wednesday's were possible.
"We saw Liverpool last night. It goes to show it's not over until it's over. Disappointed with the first leg how we started and we started like that tonight. But after the break we came out," Rose said following the Tottenham victory.
"The gaffer mentioned Liverpool's display at the hotel before the match - the gaffer doesn't mind us losing but we have to lose the right way.
"We're lucky we've gone through and we look forward to the Champions League final."
But first both sides must finish up in the Premier League, with Tottenham needing just one point at home to Everton to seal Champions League football next season.
Even with a loss, Arsenal would need to win and make up an eight-goal differential to leapfrog their London rivals.