Spurs will now switch their attention to their 2019-20 Premier League opener at home to Aston Villa next weekend, while Antonio Conte's side will continue their preparations for the new campaign against Valencia.
Tottenham needed just three minutes to make the breakthrough on home soil, with Lucas driving into the Inter box before finding the back of the net with a powerful finish.
After a slow start, Inter had a decent chance to level the scores in the 14th minute when Ivan Perisic picked out the head of Dalbert Henrique with a brilliant cross, but Hugo Lloris was on hand to make the save.
Marcelo Brozovic then sent a terrific strike against the home side's post in the 27th minute, with Inter growing into the game after struggling for possession in the early exchanges.
Spurs had the lead entering the final 15 minutes of the first period courtesy of Lucas's early effort. However, it was quite an untidy first half hour from both teams, leaving Pochettino and Conte with plenty to ponder.
A lot of the English side's good work came in the middle third of the field, while Inter were using the ball impressively without really creating many clear-cut openings.
The Italians found a leveller in the 36th minute, however, and it was a fine goal from Sensi, who collected a clever flick from 17-year-old Sebastiano Esposito before expertly picking out the bottom corner.
As it transpired, Sensi's goal proved to be the final piece of major action in the first period as English referee Andre Marriner blew his whistle with the scores level at 1-1 in London.
Both teams made a quiet start to the second half, but Inter were having joy down the left through Perisic, who has been strongly linked with a move away from San Siro during this summer's transfer window.
Three more players came off the home side's bench in the 64th minute, with Son Heung-min one of those to enter as Pochettino switched his focus to the clash against Villa next weekend.
Son had a super chance to register 10 minutes from time when he broke into a shooting position inside the Inter box, but Samir Handanovic was on hand to make an excellent save.
The sheer volume of changes from both teams in the latter stages made it difficult for either to grab a winner, and it was not much of a surprise that the contest headed for a penalty shootout.
TOTTENHAM HOTSPUR: Lloris (Gazzaniga 64'); Walker-Peters (Tanganga 80'), Sanchez (Georgiou 80'), Vertonghen (Alderweireld 46'), Rose (White 80'); Winks (Skipp 64'), Sissoko (Eriksen 46'), Ndombele (Marsh 80'); Lamela (Nkoudou 80'), Kane (Son 64'), Lucas (Parrott 80')
INTER MILAN: Handanovic; D'Ambrosio, De Vrij (Ranocchia 84'), Skriniar (Pirola 74'); Candreva (Agoume 84'), Gagliardini (Barella 46'), Brozovic (Mario 84'), Sensi (Valero 84'), Dalbert (Dimarco 74'); Esposito (Politano 46'), Perisic (Puscas 74')