The 2026 European Tour resumed at the MVM Dome on Friday, with Cross denying Greaves a triumphant debut in the Hungarian capital.
Greaves, the first woman to compete at a European Tour event, averaged 104.88 and hit five 180s.
However, Cross held his nerve in a dramatic deciding leg to progress, averaging 101.90 and hitting a 170 checkout.
“Beau is brilliant. She is going places,” Cross said after setting up a second-round meeting with Luke Woodhouse.
“She has been phenomenal on the ProTour, and I think when she learns her trade on the big stage, she’s going to do more.
“I feel really happy with my setup at the minute. I think I can do great things, but it’s down to me to prove it.”
Germany's Springer also made a winning start to his bid for back-to-back Hungarian Darts Trophy titles, beating Dutchman Maik Kuivenhoven 6-1.
“I have so many good memories coming back here,” said Springer, who will face Ryan Searle for a place in the last 16.
“I had a big smile because I was so happy to be here again, and I cannot wait to play tomorrow!”
Joe Cullen and Dave Chisnall also progressed, recording 6-3 victories over Adam Leek and Andras Borbely respectively.
Cullen averaged almost 98 and converted 60% of his attempts at double, while Chisnall hit checkouts of 101, 120 and 136 to set up a clash with Wessel Nijman.
Krzysztof Ratajski survived two match darts to beat Callan Rydz 6-5 and begin his bid for consecutive European Tour titles. He will face Gary Anderson in round two.
There were also victories for Hungarian qualifiers Laszlo Kadar and Nandor Major.
Kadar averaged more than 93 to beat Daryl Gurney on his European Tour debut and will face fourth seed Jonny Clayton.
Major produced a 124 checkout on the bull to defeat Niels Zonneveld in a deciding leg and set up a meeting with Gian van Veen.
Kevin Doets was another notable casualty, missing a remarkable 31 darts at double in a 6-4 defeat to Jonas Masalin.
Irishman Keane Barry, meanwhile, averaged almost 101 in a 6-3 victory over German number one Martin Schindler.
Tom Sykes also averaged 101 in a 6-1 win over Cameron Menzies, while Dirk van Duijvenbode beat Czechia’s David Pisek by the same scoreline.
Van Duijvenbode will face top seed Luke Humphries on Saturday evening, while James Hurrell will meet Michael van Gerwen after beating Damon Heta in a deciding leg.
Ritchie Edhouse will take on last year’s runner-up, Danny Noppert, after winning six successive legs in a 6-1 victory over Connor Scutt.
Andrew Gilding also progressed with a 6-0 win over newcomer Milan Biro, while Ryan Joyce and William O’Connor completed the lineup with victories over Cristo Reyes and Samuel Price.
Results
Afternoon session
Jonas Masalin 6-4 Kevin Doets
Andrew Gilding 6-0 Milan Biro
Ritchie Edhouse 6-1 Connor Scutt
Niko Springer 6-1 Maik Kuivenhoven
William O'Connor 6-2 Samuel Price
Keane Barry 6-2 Martin Schindler
Ryan Joyce 6-3 Cristo Reyes
Dave Chisnall 6-3 Andras Borbely
Evening session
Dirk van Duijvenbode 6-1 David Pisek
Laszlo Kadar 6-5 Daryl Gurney
Krzysztof Ratajski 6-5 Callan Rydz
Joe Cullen 6-3 Adam Leek
Rob Cross 6-5 Beau Greaves
Nandor Major 6-5 Niels Zonneveld
Tom Sykes 6-1 Cameron Menzies
James Hurrell 6-5 Damon Heta