Trump had appeared on course for a comfortable win, surging to a 12-6 lead.
However, a missed red to the middle pocket opened the door for Murphy, who seized the momentum and rattled off four consecutive frames.
Despite the pressure, Trump regrouped to clinch the 23rd frame and close out the match.
The 2019 world champion now awaits a quarter-final showdown against either reigning champion Luca Brecel or 2016 finalist Ding Junhui.
During Sunday’s session, Trump achieved a notable milestone by recording his 100th century break of the season - a feat that secured him a £100,000 bonus.
Victory at the Crucible could see the 35-year-old become the first player to surpass £2 million in a single season, with the world title carrying a £500,000 prize.
Reflecting on his performance in an interview with the BBC, Trump admitted: "Shaun was in a world of his own for four or five frames and I was a little bit worried, but I had to back myself in that last frame.
"In the past I've been a bit hard on myself but I'm a lot stronger mentally and I've learned to appreciate the losses and use them as motivation.
"I'm a lot more confident this year, I'm hitting the ball a lot better. I probably have a bit more belief in myself and I feel a lot more calm."
Si Jiahui overcomes Woollaston
Elsewhere, China’s Si Jiahui secured his spot in the quarter-finals with a 13-10 victory over England’s Ben Woollaston.
Woollaston, who had earlier knocked out fourth seed Mark Selby, fought back from 9-7 down to level at 10-10 but could not maintain his momentum as Si closed out the match with three straight frames.
Si, a semi-finalist two years ago, will next face either Ronnie O’Sullivan or Pang Junxu.