Villa had been fairly anonymous in the first half but led through Graham Burke's penalty on 50 minutes, only to see the Portuguese side equalise through Luka Stojanovic.
However, in the first minute of extra time, Irish youth international Jack Grealish used his composure to bring Callum Robinson's left-wing cross under his command and then smash the ball past Mickael Meira in goal.
Then, in stoppage time at the end of the first period of extra time, the impressive Carruthers curled home from the edge of the penalty area to seal the win.
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