The England left-back felt the Dutchman had been "rude" about Spurs players in the wake of their 1-0 first-leg defeat in north London last week.
Gullit – a former PSV star and player-manager at Chelsea – was scathing, criticising the technical abilities of the likes of Rose, Victor Wanyama and Dele Alli.
But Spurs stunned Ajax in the return fixture at the Johan Cruijff ArenA, sensationally winning 3-2 to progress to the Champions League final on away goals after a 3-3 aggregate draw – Lucas Moura scoring a hat-trick.
Mauricio Pochettino's men looked to be heading to a comprehensive defeat when Ajax went 2-0 up on the day before half-time, but Lucas netted his third in the 96th minute to complete an unlikely turnaround.
Rose was happy to hit back at Gullit following remarks he believes were disrespectful.
"Seeing people's comments like Ruud Gullit after the first game and how rude he was, it's nice we fought back and hopefully shut him up," Rose said.
"He was very critical of individual players and saying how bad we were technically, and he's bang out of order for that.
"That was fuel to the fire and hopefully he eats his words."
While happy to prove a point to Gullit, Rose was also delighted to see Tottenham deliver for manager Mauricio Pochettino.
Having seen Liverpol stage a stirring comeback against Barcelona on Tuesday evening, the Spurs boss was eager to get the message across to his players that anything is possible.
"We saw Liverpool last night," Rose told BT Sport. "It goes to show it's not over until it's over.
"There were a few words said at half-time and we came out and played like that. It just goes to show we can beat anybody when we play like that.
"The gaffer mentioned it [the Liverpool game] at the hotel before we left. He just said that's the right attitude.
"He doesn't mind losing but you have to lose in the right way. How we started against Ajax in the first game wasn't the right way.
"If we'd not won tonight he'd have been proud of the way we played at the end."