Hennepin County officials agreed Tuesday to give the proposed Southwest light-rail line an additional $20.5 million, part of a last-minute funding patch that is intended to keep the controversial $1.9 billion project afloat.

The county’s Regional Railroad Authority is already on the hook for $165.3 million for the 14.5-mile line linking downtown Minneapolis and Eden Prairie. But since legislators did not come up with their 10 percent share last spring, the Metropolitan Council is now scrambling to make up the resulting $145 million shortfall.

While voting 6-1 Tuesday in favor of the cash infusion, some Hennepin County commissioners — who sit as members of the rail authority — appeared to find their own action distasteful. Commissioner Mike Opat went so far as to call it “ulcer-inducing.”

The lone dissenter, Commissioner Jeff Johnson, questioned whether the LRT line will reduce traffic congestion as promised. He said transit advocates were “soft-pedaling a tax increase.”

But Commissioner Jan Callison offered an impassioned plea on behalf of the beleaguered project, saying it will connect people to jobs, improve the environment and bolster the region’s economic competitiveness.

“This has been a long time coming,” she said.

In a plan announced last week to rescue the line, the Met Council said it would issue $103.5 million in “certificates of participation,” an obscure financial tool similar to a revenue bond that will be used to satisfy the bulk of the project’s funding gap.

The Counties Transit Improvement Board (CTIB), a regional body that raises money mostly through a local transit sales tax, is expected to pitch in another $20.5 million.

Both the Met Council and CTIB are expected to vote Wednesday on their respective pieces of the funding puzzle. Should both bodies agree to pay more for Southwest, local funding of the project will be satisfied, enabling it to qualify for $929 million in federal matching funds.

The Met Council will apply for those funds early next year, with hopes of beginning construction the second half of 2017. The line is expected to open in 2021.

The Hennepin County rail authority is expected to raise property taxes by $3 million in 2017 to help pay for Southwest LRT and other transit projects, including the Bottineau Blue Line light-rail project, said Commissioner Peter McLaughlin.