Having lost the first leg of their quarter-final contest 3-0 at home, Juve were on the verge of forcing extra-time at the Santiago Bernabeu as they led by the same scoreline heading into stoppage time.
But then Oliver penalised Medhi Benatia for bundling Lucas Vazquez over from behind in the area and subsequently sent off Gianluigi Buffon for his reaction, which involved him putting his hands on the referee.
Cristiano Ronaldo smashed home the penalty to send Madrid through while Buffon continued his protestations and slammed Oliver after the game, with the Italian attracting strong criticism as a result.
Oliver and his wife, who is also a referee, have been subjected to abuse and threats in the wake of the match, with UEFA demanding action against those responsible.
A UEFA spokesperson said: "UEFA strongly condemns the abuse directed at Michael Oliver and his wife.
"We have been in contact with them to offer our support and we trust the relevant authorities to take action against those individuals who have behaved inappropriately both on and off social media."
Oliver is back in action as referee for the first time since that match on Monday as West Ham host Stoke City in the Premier League, having acted as a fourth official at Huddersfield Town on Saturday.