After the weekend's scoreless first leg at Riverside Stadium, Middlesbrough were beaten 2-1 after extra time on Tuesday (Wednesday AEST) by Southampton to miss out on playing Hull City at Wembley on Saturday May 23 in a game worth a minimum $250 million.
Midfielder McGree, set to be named in Tony Popovic's squad to go to the World Cup, played his part by scoring after five minutes at St Mary's Stadium.
The 27-year-old, who netted six times during a disrupted fourth season on Teesside, was substituted at halftime of extra time, while winger Silvera remained on the bench for the entire contest.
Ross Stewart drew the hosts level in first-half added time and Shea Charles' decisive goal came with four minutes left of extra time to get the Saints within one win of a Premier League return.
"I thought I put it into an area … it's gone all the way through and gone in. Incredible moment. It was a cross-shot," Charles said.
"None of us are used to playing extra time really. Really tough, against a really good team. Thankfully we came out on top."
The two games were played under a cloud, with Southampton having been charged by the English Football League with a breach of its regulations following accusations that unauthorised filming of Middlesbrough's training took place last week.
The EFL requested that an independent disciplinary commission undertake a hearing "at the earliest opportunity" but Southampton asked for more time to complete an internal review into the issue, meaning any punishment would likely be meted out before the play-off final with the Tigers.
Tensions boiled over at the end of the first half when the respective coaches — Tonda Eckert and Kim Hellberg — got in each other's faces on the touchline while being spoken to by the referee.
After the first leg, which finished 0-0, Hellberg said he "couldn't believe my eyes or ears" when he heard about the spying allegations and accused the Saints of trying to "cheat."
Following another first-half exchange — between Middlesbrough's Luke Ayling and Southampton's Taylor Harwood-Bellis — the BBC and Sky Sports reported that Ayling accused Harwood-Bellis of using discriminatory language.
Southampton are seeking an immediate return to the Premier League after relegation last season. Before that, they were in the top flight from 2012-23, while Hull City last featured in 2017.
"This is one of the toughest days in my career as a football coach, I feel for everyone around the club and I wish I could have done more," Boro boss Hellberg told BBC Radio Tees.
"Over two legs we did unbelievably and were so dominant in the first leg.
"We had to fight for the last half an hour, we had chances on the transition but we didn't get much out of them."
With AP & PA.
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