The Boston Red Sox and the Tampa Bay Rays began the week with the season’s 39th 1-0 game, but it was far different from any of the first 38. It took them 16 innings to score the game’s solitary run.

How can two major league teams play 15 or 16 innings against each other without scoring a run? There is no easy answer to that question, if any at all, at least not that the people I talked to found.

“I don’t have an answer to that question or the one about how the Red Sox fly all night, land at 6 o’clock and score 15 runs,” Gerry Hunsicker, the Rays’ No. 2 baseball executive, said, referring to Boston’s outburst in Baltimore the next night.

That the Red Sox shut out the Rays for 16 innings was not a particularly shocking event; they are tied for third in the league for having been shut out the most times (9). But for the Rays to match their scoreless innings through 15 is a different development.

Not that the Rays don’t have a good pitching staff; its 3.58 earned run average is the league’s fifth lowest. The Red Sox, however, have the A.L.’s most explosive offense. They lead the league in hitting, runs scored and on-base and slugging percentages. In a seven-day span earlier this month they scored 10 runs three times.

They have been shut out seven times, but that’s not an unusually high number. Five teams have been shut out more frequently and two others the same number of times.

It was not unusual for Boston’s Josh Beckett to hold the Rays scoreless for the first eight innings. Nor was it unusual for Jeff Niemann to prevent the Red Sox from scoring in their first eight innings.

Both are good pitchers capable of doing that sort of thing. What couldn’t have been expected, though, was the work of the relief corps, especially Tampa Bay’s.

Red Sox relievers have an aggregate 3.57 earned run average, ninth highest in the league; the Rays’ relievers’ 3.63 is 10th. Both bands of relievers, though, were superlative.

Daniel Bard, Matt Albers, Franklin Morales, Alfredo Aceves and Jonathan Papelbon pitched the second half of the game without allowing a run and giving up only three hits and a walk. They struck out seven.

Aceves, a 29-old-right-hander, who this year switched sides in the Red Sox-Yankees rivalry, pitched the 13th, 14th and 15th. He permitted no hits and walked no one but hit two consecutive batters in the 15th.

That inning was the best scoring opportunity for the Rays, whom Red Sox pitchers retired 1-2-3 in 12 of the 16 innings.

But after Aceves made contact with Evan Longoria and Casey Kotchman with one out, B. J. Upton popped out and Justin Ruggiano forced Kotcman at second.

Minutes later, in the top of the 16th, the Red Sox scored the first and only run of the game. Josh Reddick led off with a walk, Jason Varitek sacrificed him to second and Dustin Pedroia singled him home.

Adam Russell, the Rays’ eighth reliever, had just come into the game, replacing Brandon Gomes, who in three innings allowed no runs, no hits and two walks.

Cesar Ramos preceded Gomes with a 1-2-3 12th after a threatening walk-filled 11th.

J.P. Howell walked the first two Boson batters in the 11th and was replaced by Jake McGee, who walked the third batter, loading the bases. This surely would bring an end to the gamer.

McGee, however, struck out Reddick, then gave way to Joseph Cruz, who struck out Varitek and induced Marco Scutaro to pop out.

It was a remarkable effort by the Rays’ remade bullpen, the decisive run allowed by Russell notwithstanding.

The Rays willingly and unwillingly saw their 2010 bullpen evaporate, losing Rafael Soriano, Randy Choate, Joaquin Benoit, Dan Wheeler, Grant Balfour and Chad Qualls.

Who are some of these guys who have replaced them, especially the ones who made that 16-inning game as compelling as it was?

The Rays got three of the relievers in their trade of shortstop Jason Bartlett to San Diego: Gomes, Russell and Ramos. Not a bad trade when three of the players are already contributing to the major league effort.

The other two relievers who worked those extra innings, McGee and Cruz, were drafted.

Entering this season, Gomes and Cruz had played no major league baseball. Russell, McGee and Ramos had minimal major league service.

The Rays are one of the best examples of how a good organization is at least as important, if not more important than money.

These Rays, under owner Stuart Sternberg, president Matt Silverman and general manager Andrew Friedman, have no more money than the Devil Rays, under Vince Naimoli. But despite their lack of baseball experience, the Rays’ hierarchy has demonstrated an intelligence that surpasses that of many more experienced team operators.

Teams in the other divisions should be thankful that the Rays were placed with the Yankees and the Red Sox and not with them. If the Rays were in the A.L. Central, for example, their schedule would be filled with games against less potent teams against whom they could build a better record and be a stronger contender for the wild card if not a division championship.

The relief pitchers who kept that 16-inning game going are a good example of what the Rays’ braintrust can do and will continue to do. The Red Sox and the Yankees force each other to spend to keep up with or ahead of each other. The Rays have become responsible for some of that spending, too.