The Serie A leaders were 3-2 down with two minutes left but a second free-kick by Wesley Sneijder and Walter Samuel's stoppage-time winner sent the San Siro wild with delight and put the champions 11 points clear.
Injuries to a host of players, the loss of Cameroon's Samuel Eto'o to the African Nations Cup and this week's sale of Patrick Vieira to Manchester City have ravaged Inter's squad, although Mourinho was careful not to demand the club made big buys.
"If we have a few injuries, it's a problem. I had kids on the bench. I had to throw them into the fire," a breathless but thrilled Mourinho told Sky television.
"I would prefer to have a bench which gives me solutions."
Inter, who often dig themselves out of holes and did so to reach the Champions League last 16, have barely missed tall striker Zlatan Ibrahimovic since his July move to Barcelona.
Saturday's game was one occasion where a big forward would have been useful but Mourinho, who has already signed nippy striker Goran Pandev from Lazio this month, said he was fine sending defenders like Samuel up front in such situations.
Lanky forward Marko Arnautovic, on-loan from Twente Enschede, came on at half-time but has yet to impress the Portuguese, whose defence looks an even bigger problem with left backs Cristian Chivu and Davide Santon both injured.
"Technically we were lost, tactically we had great difficulty," Mourinho added.
"Siena were better than us. They were calmer, had mobility, speed, they were good on the counter-attack. They didn't deserve to lose."
Mourinho's comments did little to soothe Siena coach Alberto Malesani, who was seething about the foul awarded for Sneijder's equalising free kick.
"I speak on behalf of all the little teams who don't deserve to lose because of such banal things. That was not a foul," he grimaced.
