Lautaro Martinez's 74th minute goal sealed Inter's passage to the final of European club football's showpiece in Istanbul on June 10.
"We will only fully realise what we have done in the days to come," Inzaghi told Mediaset.
"It's a dream come true. We always believed we could do it, we have had an extraordinary journey (to the final).
"What pleased me? Everything. The running, the determination, the concentration... they were very good."
He said he could now sit back and enjoy Wednesday's second leg of the other semi-final between Manchester City and Real Madrid which is finely poised at 1-1 from the first leg.
"They're obviously two of the best teams in Europe. The first match was well-balanced.
"I'll be watching the match with a great, great deal of interest!"
Inter have been European champions three times and their last crown came in 2010 when Jose Mourinho led them to a 2-0 victory against Bayern Munich in Madrid.
