The Japan Series started in 1950, so this year was the 61st one, I suppose. And while extra-inning games themselves are not entirely rare -- they have occurred in 28 Japan Serieses -- MULTIPLE extra-inning games in the same Series are kind of rare, and full-length tie games are REALLY rare, there have only been 6 in all of Japan Series history (and 7 ties total ever).
So with a little research, I figured out the following. There have been 10 Japan Series with more than one extra-inning game. Here are those years, and I'm also including 1957 so that we get all the years that had tie games.
(I am not, by the way, including the 1953 2-2 tie game that was called due to rain after 8 innings, because this post is about LONG games. For the record, I think only two games in Japan Series history have been called due to weather; that one and the Hanshin-Lotte Game 1 in 2005 where the fog made it impossible to see anything.)
1962 : 4 (1 tie, 14 innings)
1992 : 4
1995 : 3
2010 : 3 (1 tie, 15 innings)
1950 : 2
1970 : 2
1975 : 2 (2 ties, Japan Series record - 13 and 11 innings)
1986 : 2 (1 tie, 14 innings)
1994 : 2
2003 : 2
1957 : 1 (1 tie, 10 innings)
Other years with one extra-inning game: 1953, 1955, 1958, 1959, 1961, 1965, 1966, 1969, 1973, 1974, 1976, 1977, 1980, 1983, 1988, 1996, 1997.
The tie game rules have changed a little over the years, too. Up until 1966, the rules were that extra-inning games went until sunset, since most games were played during the day until then (with the exception of the 1964 Series where the games were actually played at night). In 1967 the rule was introduced that the game length limits were no longer limited by sunset, but weren't limited by innings either -- they instead were determined by time: "you can't start a new inning after 5:30pm", or in the case of night games, 10:30pm.
After the 1986 Series went to 8 games following a Game 1 tie -- the ONLY series in history to go to 8 games -- a rule was introduced in 1987 stating that games would go to 18 innings up until Game 7, and after that there was NO limit on innings. In 1994, they instituted a 15-inning limit up until Game 7, which is the currently used limit.
The 1992 Seibu-Yakult Series was a really interesting one to reread as Yakult's 3 wins were ALL in extra innings and were all from home runs in the extra innings at that. Seibu won all of the 9-inning regular-length games and then won Game 7 as well when Yakult couldn't hit a walkoff homer, I guess.
Game 1 was tied 3-3 going into extras. Yoichi Okabayashi pitched the entire game for Yakult, but Seibu went with Yoshitaka Katori from the 10th on, and he gave up a walkoff grand slam to a pinch-hitting Toru Sugiura in the bottom of the 12th for Yakult to win 7-3.
Seibu won the next 3 games in the regulation 9 innings (with Daisuke Araki giving up a 2-run homer to Kazuhiro Kiyohara and losing Game 2, Kazuhisa Ishii losing game 3 to Takehiro Ishii, and Katori getting the revenge win in Game 4 over Okabayashi, who gave up a home run to Koji Akiyama).
So with Seibu poised to take the Series, the game moved back to Seibu Kyujo for Game 5, where it went into the 10th inning tied 6-6, and Takahiro "Bun-Bun Maru" Ikeyama hit a homer off Seibu closer Tetsuya Shiozaki to put Yakult ahead 7-6 and win the game.
Game 6 found them back at Jingu, and in this case, they found themselves in the 10th inning with the score tied 7-7. This time, the game ended when Shinji Hata hit a walkoff homer off Tetsuya Shiozaki to win the game 8-7, and tie the Series at 3 games apiece.
Game 7 featured Okabayashi facing off against Takehiro Ishii, and yet again the game went into extra innings. Seibu put up a run in the top of the 10th on a sac fly, Yakult couldn't match it, and that's how Seibu won the 1992 Japan Series.
(Hisanobu Watanabe, current Lions manager, started Games 1 and 5 and came out of neither with anything to show for it.)