Five years after Gov. Mitt Romney signed Massachusetts’ groundbreaking health care legislation, it has met its chief goal of extending insurance coverage to most residents — but with costs rising faster than inflation, lawmakers face the challenge of how to pay for it all.

Although the law has extended coverage, it has done little to fundamentally change the way consumers shop for health care, which analysts say is the only lasting solution to ballooning costs.

Massachusetts’ uninsured rate plunged to the lowest in the nation, from 6.4 percent to 1.9 percent, after the law was enacted in 2006. The rest of the nation averages close to 17 percent. Nearly every child has coverage, and more businesses are offering insurance plans...