Transport for West Midlands (TfWM) said the 700-metre stretch of track between Wolverhampton railway station and the St George's tram stop will cost an extra £10.8m – taking its total budget to almost £50m.
And bosses also revealed another delay to the Dudley transport interchange, with work now unlikely to start until "later next year" due to issues with securing land.
It marks the latest blow for the West Midlands Metro, which has undergone a miserable year marked by service suspensions, delayed works and strike action.
TfWM executive director Anne Shaw said "inflationary pressures" meant £17.8m had been added to the budgets of three key transport projects.
As well as the additional budget for the Wolverhampton extension, an extra £5m has been allocated to the delayed University station in Birmingham and £2m for the Westside Metro extension to Centenary Square.
Ms Shaw said the extra costs were largely down to increased labour costs, supply chain issues and delays to the delivery of materials.
She said cash allocated to other schemes in the £1bn Transforming Cities Fund would be used to ensure the the three projects were completed.
The Wolverhampton extension along Pipers Row is already years behind schedule and is set to finally open next spring.
Ms Shaw said "prolongation costs" were incurred due to workers having to leave the site for 18 months while the revamp of Wolverhampton Railway Station was completed.
Further delays were a result of the site having to be "cleared" during the Commonwealth Games, she said.
Ms Shaw said delays were also caused by requests to have railway lines taken out of circulation so work could take place on the Metro being either rejected or cancelled.
She also said "huge delays in the supply chain" meant that a specialist piece of high-tech kit needed for trams to operate was yet to arrive from abroad.
On the Dudley transport interchange, Ms Shaw said construction would start once all of the land needed for the scheme had been acquired through compulsory purchase orders.
"I would expect to enter into contract for construction later next year, alongside where the Metro will come into Dudley," she said.
"So the opportunity is for us to do the interchange and the Metro as one bigger project, and that will also help to manage the costs and save a little bit of money."
Ms Shaw added: "Globally, these are very challenging times and we're trying to manage all of the ambition within the resources that we have got available to us.
"It might take slightly longer and cost us a little bit more, but the ambition is still the same."
According to a new report from TfWM, in a worst case scenario the region's programme of transport projects would be £171m short.
However, this excludes the Wednesbury-Brierley Hill extension, part of which is under threat due to a funding shortfall of around £300m.
A report on the scheme, which Dudley Council said is "extremely likely" to be delayed or cancelled, is expected early next year.