Among Republican presidential candidates, it’s been demagoguery as usual. Why have a substantive debate when you can exchange inflammatory sound bites instead, especially on immigration?

But something surprising happened last week far from the campaign trail — on Capitol Hill, of all places. Just when we thought Congress would never act to address the nation’s broken immigration system, members of the House made a critical breakthrough, voting overwhelmingly to approve a fix that will make American companies more competitive and the immigration system fairer and more welcoming.

The Fairness for High-Skilled Immigrants Act is not a broad, far-reaching bill. It’s nothing like the comprehensive overhaul Congress has been debating for years. But this small change could have significant consequences, for thousands of immigrants and for the politics of immigration.

The per-country caps that would be eliminated by the bill are among the most absurd and cumbersome features of the U.S. immigration system. Under current law, thousands of foreigners are approved each year to enter the United States as legal permanent residents, some sponsored by employers who need their skills, others by family members who arrived before them and became citizens.

But this approval is not enough to guarantee a visa. Instead, approved candidates get in line and wait for their number to come up under the annual cap for their countries. And because until now visas were allotted equally to all countries, no matter how big or small, candidates from big countries with strong ties to the United States have often waited years. Backlogs have gotten so bad that workers from India, for example, currently face waits of 70 years — in other words, many never get visas — and family members from Mexico wait more than a decade.

Phasing out the caps would dramatically reduce waits for many of the highly skilled workers that America needs to remain a globally competitive knowledge economy. American companies will find it easier to hire researchers, engineers and other top talent from the big countries that produce most of the brainpower they rely on to do business. The United States will become a more-attractive destination for foreign innovators and entrepreneurs. And they in turn will help create jobs for Americans, a much-needed boost for economic recovery.

Many immigration reformers wish the measure went further, not just easing and eliminating country caps but also expanding the number of legal permanent residence permits, or green cards, issued every year. Unless we do this too, the critics argue, easing the caps is no more than a palliative. It robs slots from some countries and awards them to others, shortening waits for Indians and Mexicans but lengthening them for immigrants from other, smaller countries.

The critics aren’t wrong: We do need more green cards. But that isn’t politically possible today, not with unemployment hovering above 8 percent and politicians so gun-shy about immigration.

Still, the House bill represents a huge political breakthrough. Lawmakers in Washington have been trying to fix the system for a decade. Democrats and Republicans have been unable to agree on an approach. The politics of immigration have grown so polarized that it sometimes seems Democrats own the issue, while few Republicans will touch it.

The process that produced the Fairness for High-Skilled Immigrants Act was very different. The bill was proposed by Republicans. With some unusually nimble negotiating, its GOP sponsors persuaded influential pro-immigration Democrats to work with them. The resulting bipartisan measure was approved overwhelmingly, by a vote of 389 to 15. And although it has encountered some obstacles in the Senate, it has broad bipartisan support there as well.

Perhaps most important, the bill could usher in a new approach to immigration reform. For a decade, the conventional wisdom among reformers has been “all or nothing.” We shouldn’t pass any small fixes, the thinking went, because that would take the pressure off — we’d lose the head of steam we needed to pass comprehensive reform. But this strategy has produced only stalemate.

So maybe it’s time to try a different approach: not all or nothing but step by step.

Tamar Jacoby, a fellow at the New America Foundation, is president of ImmigrationWorks USA, a national federation of small-business owners.