Spurs got the better of Bayern Munich on penalties on Wednesday to win the Audi Cup, but they were second best in Sunday's International Champions Cup clash against Antonio Conte's side.
Inter pegged the hosts back before half-time through new signing Stefano Sensi after Lucas Moura had maintained his impressive form by putting Spurs in front inside the first three minutes.
Neither team could find a winner in a match lacking clear-cut chances and it was Inter – still three weeks away from their first competitive match of 2019-20 – who prevailed 4-3 on penalties in London, Joao Mario converting the winning kick.
The Brazilian forward was picked out by Erik Lamela on the counter, carried the ball into the opposition box and fired his shot past Samir Handanovic from 12 yards.
Marcelo Brozovic came close to responding for Inter with a powerful shot from range that crashed off the inside of the post and rolled across goal.
But the visitors did not have to wait much longer for an equaliser, as Sensi kept his composure after being played in behind by Sebastiano Esposito and tucked the ball past Hugo Lloris in the 36th minute.
The second period was broken up by a flurry of substitutions and it took until the 77th minute for the first on-target attempt of the half – Nicolo Barella testing Lloris' replacement Paulo Gazzaniga from distance.
Son Heung-min was then denied by Handanovic from 10 yards as the match opened up, but penalties were required to separate the sides.
Eriksen was thwarted by Handanovic from Tottenham's first attempt, only for George Puscas to meet his match in Gazzaniga.
The next six penalties were converted, but Skipp could not find a way past Handanovic and ex-West Ham man Joao Mario secured the win.