Indian cricket team (men) has many test match wins outside Asia. And many of them, in particular against Australia and England, are absolutely thrilling.
Personally, one of the most memorable came in December 2003 when India was touring Australia. It was the second test match in Adelaide when Rahul Dravid
pulled one of the best test performances ever, scoring 233 runs; in the second inning, he scored 72 runs. This match cemented his legacy as one of the greatest in the game.
played equally well, scoring 148 runs, helping Dravid.
(Courtesy: Zee News)
In the same match, while Dravid and Laxman were leading with the bat, Ajit Agarkar was a complete magic with the ball. He took 6 wickets, conceding only 41 runs.
India won that match with 4 wickets remaining.
I believe, among many, this was one of Indian cricket team
's (men) best and most exciting test match wins outside Asia.
(That series was also last for Australian captain Steve Waugh, which made things even more memorable!)
The second test match between India and West Indies in March 1971 at Queen's Park Oval is also often hailed as one of team's best test match wins overseas. At the time, West Indies players were the beast in the game, dominating the #1 position.
It was the debut match for legendary Sunil Gavaskar. While the hosts were expecting to win the match rather easily, a handful of Indian players rose to change the tide. Dilip Sardesai, Gavaskar and Eknath Solkar took charge with bat. In the bowling department, Venkata Raghvan and Bishan Singh Bedi shone brightly.
(Courtesy: The Hindu)
That match ended on a draw. However, Indian Cricket Team put up a great fighting spirit that won hearts. India won that series with 1-0, which put our country on the global platform in the cricketing world.
Aside from these two, I am sure there are many test match wins for the Indian team outside Asia. Which ones are best and most exciting, it really depends on personal preference and interests.
Hope this answers your question well!