The Italian side had Francesco Toldo to thank for an inspired goalkeeping display that enabled his side weather a storm of Valencia attacks and take it into the last four for the first time in more than 20 years.
Hector Cuper's team will now face the winners of Wednesday's quarter-final between neighbours AC Milan and Ajax of Amsterdam.
Valencia, who lost 1-0 in the first leg at the San Siro, made an ambitious start, pouring forward in search of an early goal to level the tie.
Midfielder Miguel Angel Angulo went close with a rising 30-metre drive in the third minute, but just as Valencia, who reached the final in their last two Champions League appearances, looked set to take control, Inter made the most of an uncharacteristic error by centre back Roberto Ayala to snatch the lead.
The Argentinian international failed to cut out a simple through ball from the Inter midfield on four minutes and it squirted free into the path of Vieri who sprinted clear towards goal.
Spanish 'keeper Santiago Canizares produced a superb reflex save to get his hand to Vieri's first shot, but he was powerless to stop the Italian from snapping up the rebound and poking home into the open net.
Vieri later hobbled off in the 32nd minute with what was thought to be a knee ligament injury and was replaced by Uruguayan striker Alvaro Recoba.
Inter barely had time to celebrate their early lead before the Spaniards netted a beautifully crafted equaliser in the seventh minute.
Receiving the ball with his back to goal on the edge of the area, Norwegian striker John Carew turned defender Ivan Cordoba before slipping the ball square to Pablo Aimar who side-stepped Carlos Gamarra and drilled a low shot past Toldo into the far corner.
Valencia went on to subject Inter to a period of intense pressure with right-back Anthony Reveillere producing a succession of dangerous crosses from the right and Vicente Rodriguez following suit on the left.
But a combination of some desperate last-ditch defending and some brilliant goalkeeping from Toldo kept the Spaniards at bay.
The Inter 'keeper parried a testing Aimar free kick on 21 minutes and then pulled off a superb one-handed save to deny Reveillere soon after.
The Italian international went on to produce a succession of three brilliant reflex saves in the 40th minute to parry a double from Rodriguez and a follow-up from Reveillere.
There was no respite in the relentless Valencia pressure after the break and Toldo pulled off yet another spectacular save to turn aside a goal-bound volley from Ruben Baraja in the 50th minute.
But he could do nothing to stop the midfielder from putting the home side 2-1 ahead with a diving header from the resulting corner.
The 50,000 Mestalla crowd went wild a couple of minutes later when defender Marco Materazzi appeared to bring Juan Sanchez down as the pocket-sized forward darted into the Inter area, but Danish referee Kim Milton Nielsen waved play on.
Inter were rarely able to venture into Valencia territory and their only real chance of the half came 10 minutes from time when Canizares got his foot to a sharp shot from Recoba.
