After drawing 2-2 in the first leg at Ibrox last week, the Light Blues were put to the sword in the Philips Stadion, starting in the 35th minute when Ismael Saibari headed in the opener.
The Moroccan attacker added a second in the 53rd minute and while Gers skipper James Tavernier pulled a goal back in the 64th minute, PSV captain Luuk de Jong soon restored the two-goal lead before a Joey Veerman strike and an own goal by Gers defender Connor Goldson sealed a miserable night for the Light Blues, who drop down to the Europa League.
Beale, who has to turn his attention quickly to the visit of Celtic in the cinch Premiership on Sunday, told broadcasters TNT: “Obviously hugely disappointed with the result tonight.
“I felt over the two legs we’ve struggled to handle De Jong and Saibari, the two forwards in both legs and in key moments in the game, certainly in set plays, we conceded.
“Listen, the buck stops with me. I wasn’t happy with losing the way that we lost tonight.
“There was moments in the game that could have gone for us, I thought in the second half we had some other opportunities didn’t go for us, but it’s obviously bitterly frustrating.
“In two legs against PSV, we scored three goals, the problem’s at the other end of the pitch.
“I have to say they’re a very good team but that doesn’t take away from our disappointment.
“Coming into the September international break, the aim was obviously to get in the Champions League, it was to get into the quarter-finals of the Viaplay Cup and it was to end this first period of league fixtures as high up or in front if we can. That’s still in our hands.
“Today is obviously bitterly disappointing. We’re out of the Champions League. I think we played against an excellent team and just an apology to our fans who came across. They stuck with the team in both legs and they expect a little bit more than what they got.”
Tavernier insists Rangers have to “move on” quickly, saying: “It is obviously really raw and hard to take.
“We knew how much it meant to the fans and the club and the scoreline is really disappointing.
“But it’s something that we obviously have to move quickly on because we’ve got a really important league game at the weekend.”