Planning permission has already been granted for the ambitious new centre, which is due for completion late next year.
The latest funding win comes from Arts Council England, and will be supported by funding already committed by the Garfield Weston Foundation and The Foyle Foundation.
An on-going fundraising campaign will raise the remaining £1.7m still needed to deliver the project at the West Bretton park, near Wakefield.
Designed by London-based architects Feilden Fowles, recently named BD Young Architect of the Year 2016, the centre will be placed at the southern entrance to the Park, less than a mile away from junction 38 of the M1.
It has been designed to make minimum impact on the site and to fit sympathetically with the historic landscape.
It will include a restaurant, gallery space, a public foyer and a shop.
Michelle Dickson, Director North, Arts Council England, said: “Yorkshire Sculpture Park continues to go from strength to strength, and makes an outstanding contribution to the visual arts in England. With visitor numbers growing year on year, the development of its new visitor centre will provide improved facilities for the public, and more exhibition space.
“I’m delighted that we’ve been able to help support this project and look forward to seeing its development.”
Founder and executive director of YSP, Peter Murray said: “We are delighted to have the backing of Arts Council England. The new visitor centre is a reflection of our ambition to increase long-term resilience and sustainability by building audiences, further developing our artistic programme, and increasing visitor income.
“In our 40th anniversary year, the centre will provide an important new element to our physical infrastructure, bringing together all of the successful elements of previous developments. It will provide a platform to sustain and increase visitor numbers over the next decade, offering exciting new artistic experiences for the public to enjoy, whilst boosting our commercial income, providing sustainability in the long term as reductions in public funding continue to take effect.”