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The Yankees' bullpen, as currently constructed, has some impressive pieces, and an ability to shut teams down. For a variety of reasons, they will never put together as impressive an outing as the bullpen, or rather one part of the bullpen, did on August 25, 1976.
The Yankees were comfortably in first place on August 25, 1976. On that day, they were wrapping up a series in New York against the Minnesota Twins. Ed Figueroa got the start for the Yankees that day. Figueroa finished fourth in Cy Young voting that season, but this day wasn't his best.
After a scoreless first inning, the Twins racked up four runs on Figueroa in the second. Two errors and four hits in the inning put the Yankees in an early hole. They got one run back in the bottom of the second when Graig Nettles homered.
Three innings later, as Figueroa settled down, the Yankees came back to tie the game. Back-to-back hits from Roy White and Thurman Munson with runners in scoring position tied the game at four. The Yankees missed a chance to take the lead an inning later when they stranded runners at second and third.
Having settled down somewhat, Figueroa made it through six innings and came back out for the seventh. After allowing a double and then getting a fly out, Figueroa was taken out. Billy Martin opted to bring in Dick Tidrow. Tidrow had made 25 starts for the Yankees in 1974, but had spent '75 and '76 mostly working out of the bullpen. He stranded the Twins' runner in the seventh, keeping the game tied at four.
After that neither teams' offense could push any runs across. Thankfully for the Yankees, Tidrow was doing a good job in long relief. In this case, it was very long relief. On August 25, 1976, Dick Tidrow pitched 10.2 innings out of the bullpen. He allowed just four hits and didn't allow a run. After allowing a lead-off single to Steve Byre to start the 18th, he was finally replaced.
Grant Jackson was brought in to relieve Tidrow. He got out of the 18th and then threw a 1-2-3 inning in the 19th. Oscar Gamble led off the bottom of the 19th with a single. After a bunt moved him into scoring position, he eventually scored when Mickey Rivers singled.
Dick Tidrow's relief outing is not the longest in Yankees' history. It is the longest scoreless relief outing in Yankees' history, however. And given the way that bullpens are used today, I don't think it's a record that is in any danger of being broken.
Sources
http://www.baseball-reference.com/boxes/NYA/NYA197608250.shtml