I always thought that those types of matches rarely happen (excluding friendlies, unofficial football matches/competitions and contests over two legs), but after doing a thorough search on the internet I found a fair amount of matches and I arranged them according to date.
What's more impressive that most of these matches happened within the last 20 years. This long list might be incomplete, so anyone is more than welcome to add any additional matches that I have might missed.
1) Charlton Athletic F.C. 7–6 Huddersfield Town F.C. in Football League Second Division 1957–58. Huddersfield were leading 5–1 with just 27 minutes remaining, but Charlton was able to win the game eventually!!
2) Portugal 5-3 North Korea in quarterfinal round of World Cup 1966. The North Koreans scored three unanswered goals in the first 24 minutes. Portugal needed a four-goal effort by Eusébio to pull out a 5-3 victory.
3) Bayern München 6 -5 VfL Bochum in Bundesliga 1976. Bochum were 3-0 up by the half-time whistle and 4-0 in 53th minute. In the next 20 minutes, Karl-Heinz Rummenigge and Gerd Muller took Bayern to an unbelievable 5-4 lead. Kaczor put Bochum level on 5-5 in the 80th minute and that seemed to be it, but with a minute to go Ulrich Höness scored his second and Bayern's sixth for their most amazing win ever.
4) Nigeria 4-4 Russia in FIFA World Youth Championship 1989. Russia was leading 4-0 after 58 minutes into the match, but in the space of 30 minutes, Nigeria fought back to equalize 4-4 and take the match into penalty shoot-out. The Nigerians triumph 5-3 in penalties.
5) Werder Bremen 5-3 Anderlecht in UEFA Champions League 1993. The Belgians led 3-0 until the 66th minute, where the reigning Bundesliga champions Werder was able to score 5 goals within 23 minutes.
6) Liverpool 3-3 Manchester United in the English Premier League 1994. United raced into a three goal lead within the opening 24 minutes. Goals from Denis Irwin, Steve Bruce and Ryan Giggs had silenced the Kop and it looked for a little while like it could turn into a rout. But two goals from £2.3million signing Nigel Clough and a header from Neil Ruddock completed the comeback with 11 minutes to go and sent Anfield wild.
7) DR Congo 4-4 Burkina Faso in African Cup of Nations 1998. Burkina Faso was leading 4-1 until the 86th minute, but Congo was able to draw the game with 3 goals in the last 4 minutes!!!
8) Manchester United 5-3 Tottenham Spurs in the English Premier League 2001. Spurs were 3-0 up and coasting until the end of the first half, with goals from Dean Richards, Les Ferdinand and Christian Ziege. Man U launched an astonishing 5 goals comeback in the second half through Cole, Blanc, van Nistelrooy, Veron and Beckham.
9) Cameroon 5-5 Portugal in FIFA U-17 World Championship 2003. Portugal was leading 5-0 until the 70th minute, where Cameron was able to score 5 goals within the last 20 minutes!!
10) Manchester City 4-3 Tottenham Spurs in FA Cup Fourth round 2004. Spurs took a convincing 3–0 lead into the break after goals from Ledley King, Robbie Keane and a Christian Ziege free kick. However, after the break, goals from Sylvain Distin, Paul Bosvelt, Shaun Wright-Phillips and Jon Macken in the 90th minute sealed a victory, attributed to the "magic of the FA Cup."
11) Liverpool 3-3 A.C. Milan in the final of the UEFA Champions League Final 2005. The most famous one here, for sure. Milan were regarded as favourites before the match and took the lead within the first minute through captain Paolo Maldini. Milan striker Hernán Crespo added two more goals before half-time to make it 3–0. In the second half Liverpool launched a comeback and scored three goals in a dramatic six-minute spell to level the scores at 3–3, with goals from Steven Gerrard, Vladimír Šmicer and Xabi Alonso. Liverpool won eventually through penalty shoot-out 3-2.
12) Peterborough 4-4 Cardiff in the English Football League Championship 2009. Cardiff was leading 4-0 in the 1st half, however Peterborough was able to draw with another 4 goals in the second half.
13) Mali 4-4 Angola in Africa Cup of Nations (CAN), 10 January 2010. Mali pulled off a stunning fightback to draw 4-4 with Angola after trailing 4-0 with 11 minutes left in the opening game of the Africa Cup of Nations.
14) Motherwell 6–6 Hibernian in the Scottish Premier League, 5 May 2010. Motherwell came from 6-2 down after 65 minutes into the game to draw 6-6 with Hibernian. Motherwell even missed a penalty kick in the process. The last astonishing goal came in the 93 minute.
15) Preston North End 6-4 Leeds United in the English Football League Championship, 28 September 2010. Preston fought back from 4-1 down after 39 minutes into the game to beat Leeds 6-4 at the end of the game.
16) Newcastle 4-4 Arsenal in the English Premier League, 5 February 2011. Arsneal was leading 4-0 just after 26 minutes into the match, until Newcastle was able to draw with another 4 goals starting from the 68th minute.
17) A.C. Milan 4-3 Lecce in the Italian league Serie A, 23 October 2011. AC Milan produced an excellent second half comeback, coming from 3 goals down to win 4-3.
18) Manchester United 3-3 Chelsea in the English Premier League, 5 February 2012. Sir Alex Ferguson's team were 3-0 down after 50 minutes, however Wayne Rooney (2 penalty goals) and Javier Hernandez lead astounding comeback for Manchester United.
19) Sweden 4-4 Germany in World Cup 2014 Qualifiers for Europe, 16 October 2012. Sweden staged an extraordinary comeback in their World Cup qualifier with Germany, fighting back from 4-0 down with 30 minutes left to draw 4-4.
http://www.youtube.com/watch?v=_IQ6KOG4TEk (Swedish commentary)
http://www.youtube.com/watch?v=fDrlpkHqX2Q (German commentary)
20) Arsneal 7-5 Reading in Football League Cup (Capital One Cup) quarter-finals, 30 October 2012. With Arsenal 4-0 down after 37 minutes, the Gunners recovered with 4-4 tie in the last minute of the second half, leading to the 3rd and 4th half. Reading scored its 5th goal in the 115 minute after Arsenal was leading 5-4 in the 103 minute, however Walcott and Chamakh added 2 more goals in the last 3 minutes of the game.
21) West Bromwich 5-5 Manchester United in the English Premier league, 19 May 2013. West Brom was falling behind twice by 3 goal difference in the game (3-0 and 5-2), but Lukaku's late double and Youssouf Mulumbu denied Ferguson a winning end in his 1,500th and final game in charge of Manchester United.
22) Colombia 3-3 Chile in World Cup 2014 Qualifiers for South America, 11 October 2013. Chile was leading 3-0 after just 30 minutes into the game, But a red card to Chilean midfielder Carlos Carmona in the 66th minute allowed Colombia to recover with Teofilo Gutierrez scoring the home side's first three minutes later before James Rodriguez was fouled in the box on two occasions and, both times, Falcao converted from the spot.