We've had two asia cup of test matches as well, won by Pakistan in year 1998 and Sri Lanka in 2001.
Here's a brief history :-
April 1984 in SharjahIndia v Sri Lanka v Pakistan : won by India.
Only 3 match triangular, each team played other once, and India (who won both games) were declared winners without there being a final!
Mar-Apr 1986 in Sri LankaPakistan v Bangladesh v Sri Lanka : won by Sri Lanka.
India skipped due to strained relationship with Sri Lanka, Bangladesh joined this second edition of Asia cup cricket which was another triangular tournament (each match of 45 overs) but with a final, between Sri Lanka & Pakistan which hosts won by 5 wickets.
Oct-Nov 1988 in BangladeshIndia v Pakistan v Sri Lanka v Bangladesh : won by India.
A tournament dominated by finalists India and Sri Lanka who both team Bangladesh and Pakistan at league stage but Indians had a better day on final, winning by 6 wickets after bowling out islanders for only 176.
December 1990 in IndiaIndia v Sri Lanka v Bangladesh : won by India
Pakistan refused to tour India and hence only 3 teams played, two major ones reached final at Kolkata in which India yet again beat Sri Lanka, this time chasing down 204 with 7 wicket and 17 balls to spare.
April 1995 in SharjahIndia v Sri Lanka v Pakistan v Bangladesh : won by India
Asia cup returned after 5 years (after proposed 1993 edition was canceled due to ill political relationship between India & Pakistan), back to where it all started at Sharjah. Pakistan defeated India by huge margin of 97 runs but lost their way since then, losing to Sri Lanka and failing to reach finals where India, having beaten Bangladesh and Sri Lanka at league stage, beat Lankans once again to claim their 3rd consecutive Asia cup, chasing down 232 with 8 wickets in hand.
July 1997 in Sri LankaSri Lanka v Bangladesh v Pakistan v India : won by Sri Lanka.
Crucial India vs Pakistan match was rained off despite there being a rest day, which cost green team a chance for finals as Indians punished Bangladesh within 15 overs in last league game to improve their net runrate and made it finals against hosts Sri Lanka who chased down 240 to win their second asia cup.
May-June 2000 in BangladeshIndia v Pakistan v Sri Lanka v Bangladesh : won by Pakistan.
An undefeated tournament for Pakistan who won asia cup for the first time, beating Bangladesh by 233 runs, India by 44 runs and Sri Lanka by 7 wickets and again by 39 runs in finals at Bangabandhu stadium.
July 2004 in Sri LankaIndia v Pakistan v Sri Lanka v Bangladesh v Hong Kong v UAE : won by Sri Lanka.
Minnows Hong Kong and United Arab Emirates were included with Bangladesh joining seniors India, Pakistan and Sri Lanka in this 8th edition of asia cup. A 12 match tournament, India and Sri Lanka played finals where hosts defended smallish 228, winning by 25 runs.
June 2008 in PakistanIndia v Pakistan v Sri Lanka v Bangladesh v Hong Kong v UAE : won by Sri Lanka.
Another 6 team 6 league game 6 super 8s game tournament but a very high scoring one where totals over 300 were chased down frequely, and Indians yet again made it finals with Sri Lanka in opposition only to lose it by 100 runs.
June 2010 in Sri LankaIndia v Sri Lanka v Pakistan v Bangladesh in Sri Lanka : won by India.
Sri Lankans survived Shahid Afridi blitz in first match while Indians escaped Bangladesh fear coming into the tournament after a gap. Lankans shut Bangladesh down with bonus point to book their final berth while India snatched a thriller against arch rivals Pakistan off second last ball. Indians lost to Lankans in dress rehearsal but in final, they brought their A game to win convincingly.
March 2012 in BangladeshIndia v Sri Lanka v Pakistan v Bangladesh in Bangladesh : won by Pakistan.
Easily the best performance from Bangladesh in any tournament thus far, home team not just beat India and Sri Lanka but also ran Pakistan utterly close in both the encounters but fell 21 runs short in league match and then 2 runs short in final. Indians lost just one match and that was enough for them to return home early while Sri Lankans, tired, didn't win a single match. Pakistan claimed the tournament in a tense finish.
March 2014 in BangladeshIndia v Sri Lanka v Pakistan v Afghanistan v Bangladesh in Bangladesh : won by Sri Lanka.
Twice before, in Sri Lanka in 1997 and Pakistan in 2000, had won Asia cup undefeated but with 4 wins. In this 2014 edition, Sri lankans walked away with 5 comprehensive wins, beating Pakistan in the final who had progressed ahead of Indians after a thrilling encounter. New commers Afghanistan impressed too, beating home team Bangladesh who failed to win even a single match.