Manager Paolo Di Canio quit on Monday citing off-field problems at the club and Piccareta revealed after the victory at Prenton Park that the backroom staff would follow suit.
Despite the turmoil the Robins moved into pole position in the league after James Collins' strike and two fine goals from Gary Roberts condemned Tranmere to a fourth consecutive defeat. But Piccareta, who was put in charge for the trip to Tranmere, was adamant it would be his final game.
"After this game, this fantastic game that everyone has witnessed, I want to tell you that this is our last game as technical staff of this football club," said Piccareta. "I want to highlight one thing. This achievement that this fantastic team got, so top of the league and this victory has a name, and a surname and that is Paolo Di Canio.
"People must know that (on Monday) he released a statement at 7pm, but until 5pm we were together preparing for this game as usual, in the best professional way. And I just spoke to Paolo because he just rang me to congratulate with ourselves and with the players because he is absolutely happy of what his team has achieved.
"From (Wednesday), I don't know what is going on with Swindon Town Football Club but this was a very important moment to say thanks to the Swindon Town fans that are amazing.
"In the morning we will resign because we are here for Paolo Di Canio and we will follow Paolo Di Canio everywhere. Even if he will coach Real Madrid or Luton Town."
Tranmere boss Ronnie Moore blamed his side's decision making for the defeat and said: "It's self destruction. We've lost the game because we've made silly wrong decisions.
"The goalkeeper makes a mistake for the first then he gives them the third with a mis-kick straight to the lad who to be fair smashes it from 50-60 yards and it goes in. It's frustrating for everybody, there's no doubt about that.
"It's serious for us. We need to get back to winning ways. One or two in the crowd were having a go and you can understand why. They are frustrated. But what they've got to realise is that that team that's played here is seven players short of what the team has been."