Aston Villa can handle the pressure of a relegation battle and have given themselves a chance of staying up by beating Crystal Palace, says boss Dean Smith.
Trezeguet scored twice as struggling Villa secured a first league win since 21 January to move to within four points of safety with three games to go.
"We have found it hard this season and there is no getting away from that," said Smith after his side's first win in 11 league games.
"I know these players can play under pressure. We have been playing tournament football since we came back.
"We haven't got the points we need but we're still in the tournament."
In an eventful game at Villa Park, Mamadou Sakho thought he had put Palace ahead when the ball appeared to strike the defender's shoulder on its way into the net - but it was ruled out for handball after a lengthy delay, much to the visitors' dismay.
Villa then took the lead just before half-time when Egypt winger Trezeguet side-footed the ball into the net at the back post after Palace had failed to clear a free-kick.
Shortly after the break, the hosts were awarded a penalty by referee Martin Atkinson after Patrick van Aanholt's challenge on Jack Grealish, but the decision was overturned after another long VAR check.
Villa sealed the points when Trezeguet toe-poked home his second goal and, after the final whistle, Palace forward Christian Benteke, who was playing against his former club, was shown a red card as tempers boiled over.
Palace, who have now lost their past five league games, started with Wilfried Zaha, who revealed before the match he had received racist abuse on social media.
Palace manager Roy Hodgson described the abuse of the 27-year-old Ivory Coast forward as "cowardly and despicable".
Follow all of Sunday's Premier League actionRacist abuse of Zaha 'cowardly and despicable' - HodgsonHuge three points for Villa
Villa are running out of time to pull off a miraculous escape but a battling win at least gives them a fighting chance of avoiding the drop.
After victories for fellow strugglers Watford and West Ham United on Saturday, the visit of Palace was a must-win game for Villa and Smith's side delivered - even if they did require a slice of luck to achieve it.
With the game goalless, it appeared a harsh VAR decision to deny Sakho a goal because of handball as Luka Milivojevic's looping ball appeared to come off the defender's shoulder on its way past keeper Pepe Reina.
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Villa recovered from that scare to seize control, Mbwana Samatta heading over a good chance before Grealish was denied by a smart Vicente Guaita save.
The breakthrough came on the stroke of half-time, Trezeguet pouncing after Conor Hourihane's free-kick was inadvertently flicked on by Joel Ward into the path of the forward.
Villa, who were forced into a late change after losing defender Kortney Hause to a groin injury in the warm-up before the game, kept their focus after having Atkinson's decision to award a penalty overturned, and Trezeguet struck a crucial second after the ball fell to him inside the penalty area.
Reina produced a fine stop to keep out Van Aanholt soon after Guaita had prevented Hourihane from making it 3-0.
Villa's aim is to be in with a chance of staying up by the time they face West Ham on the final day of the season on 26 July. This result at least gives them hope.
Benteke 'did something he is now going to regret'
Palace will be upset that Sakho's goal was overturned but they were well beaten in a match which ended in ugly fashion.
Benteke, who was twice denied in the first half by Reina, was shown a red card following a flashpoint after the final whistle had been blown.
The forward appeared to kick out at an opponent and now faces a three-match suspension which will rule him out for the remainder of the season.
It completed a miserable day for Hodgson's side, who have lost five of their six games since the Premier League resumed.
"Christian Benteke has missed a lot of football this season through injury and got himself fit to play," said Hodgson.
"Now he has taken himself out of the last three of them for kicking out at an opponent. There is nothing I can say to excuse it. I'm obviously very unhappy about him being sent off.
"Quite frankly I don't know what he is going to be able to say to any of us to justify it other than the fact the red mist came down. He did something he is now going to regret."
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Palace, who have already secured an eighth successive campaign in the top flight, have failed to score in five of the past six matches.
With home games against Manchester United and Tottenham, as well as a trip to Wolves, still to come, Hodgson will be seeking an improvement from his side before the end of the season.
Man of the match (Trezeguet - Aston Villa)
Back in the goals - the stats
Villa scored as many goals in this win as they had across their previous eight league matches combined beforehand (two).Palace have lost their past five league games, their poorest run since they lost eight in a row in September 2017.Palace have failed to score in the first half of 26 of their 35 Premier League games this season - more than any other side.Trezeguet scored his fourth and fifth Premier League goals for Villa, and his first since December 2019 against Chelsea.Only Jack Grealish (seven) has scored more Premier League goals for Villa this season than Trezeguet (five), who became only the second Villa player to score a double in this season's competition (also Wesley against Norwich in October 2019).What's next?
Both teams are back in action on Thursday. Villa travel to Merseyside to face Everton (18:00 BST), while Crystal Palace host Manchester United at Selhurst Park (20:15).