CHICAGO -- That new White Sox concept where a fortified bullpen can compensate for a lack of offense is going to have to take some time to develop.

Unable to add an impact hitter at the trade deadline, the White Sox went for bullpen help instead last week when they added Jason Frasor from the Toronto Blue Jays.

They might not be scoring runs, the move seemed to suggest, but they will take the runs they do get on a given day and make the most of it with an improved pitching staff.

It seemed to work just fine in a low-scoring victory Friday against the Red Sox, but on Sunday the plan backfired. With Jesse Crain called on in the seventh inning and the White Sox clinging to a one-run lead in yet another low-scoring game, the Red Sox rose up for two runs on a single by Dustin Pedroia.

The Red Sox then added an insurance run in the ninth inning that was charged to Frasor. Will Ohman gave up an RBI double to Adrian Gonzalez that allowed one of Frasor’s runners to score.

“Every time the manager makes a move and it doesn't work, it seems like a bigger mistake because we don't have that many runs on the board,” manager Ozzie Guillen said. “We keep waiting and waiting and waiting to see if we can't have that space. We get a big base hit, a bases-loaded double or something like that, just to have some space to make mistakes. Unfortunately, that did not happen.”

Improvement in the close games could go a long way to helping the cause. The White Sox entered play Sunday with a perfectly mediocre 15-15 record in one-run games and an 11-11 mark in two-run games. Even in three-run games they were 8-9.

The numbers illustrate the White Sox’s mediocrity this season. Since Frasor has been on board, the White Sox are 1-1 in two-run games so the trend continues.

“When you don't score runs in the American League, it's going to be hard to win games,” Guillen said. “The pitchers can take you so far. Every game is game is a battle. Every game, know not that one mistake can cost you the game, since [opponents] can take the lead.”

The shame of it, of course, is that a group that had a 1.65 ERA over its past 27 games and a 1.70 ERA over its past 35 games gets put under the microscope. The bullpen has done its job this season, and it makes no sense to hang Sunday’s defeat on them.

Instead, it’s the inconsistent offense that has caused problems. Early in Sunday’s game, the White Sox appeared to be getting the hits they needed with RBIs from Brent Morel, Alex Rios and Alexei Ramirez. After Ramirez’s RBI single in the fourth inning, though, the White Sox loaded the bases with one out but didn’t score when Carlos Quentin struck out looking and Adam Dunn popped out.

"Every time you go out there, you just try to throw up zeroes and give our offense a chance to score," said White Sox starter Mark Buehrle, who gave up two runs in six innings Sunday. "Obviously, when we’re struggling we try to go out there and say 'I can’t make too many mistakes.' If we get down by too much, it’s going to be harder to come back."

It's a scenario White Sox pitchers have faced often this season.

“That's White Sox baseball,” an exasperated Guillen said. “I don't think we've had a game where we can go out there and relax a little bit, coaching staff or pitching.

“Every game it seems like we're up by one, down by two, up by two. It's close games every day. We try to bear down and sometimes, we cannot make one mistake as a pitcher because every time we make a mistake, you're going to pay.”