Murphy, who won the UK title in 2008, came through his clash with Graeme Dott 6-2, while newly-crowned Premier League Snooker champion Bingham downed Stephen Maguire 6-4.
Murphy will next play either Mark King or young Belgian Luca Brecel; Bingham will meet Ali Carter in his quarter-final.
In the battle of the former world champions, both Murphy and Dott had several bites at the cherry to get something going before Murphy eventually edged a scrappy opener. A quickfire 28 break got Murphy off and running in the second and a bad miss on the pink to the middle followed by a wild attempted red to the corner allowed Murphy to establish a two-frame advantage.
A missed black off its spot from Murphy let Dott in for a 31 break but the 'Pocket Dynamo' again failed to post a big enough total to put Murphy away and a clearance of 62 put the former champion even further ahead despite not playing his best snooker.
Breaks of 25 and 54 in the next finally got Dott on the scoreboard at 3-1 but another loose pot attempt let Murphy in for a superb 130 clearance to pile further misery on the Scot. The sixth looked to be following a similar pattern with Murphy in and around the black with his break on 40. But a missed red to the corner let in Dott for a frame winning 71 to get back to within two.
Dott was in first in the next but an unlucky split on the reds let Murphy back to the table and he took full advantage with a break of 75 to put him within a frame of the quarters. And the last of a number of misjudged pot attempts proved the final straw for Dott, Murphy banging in another frame wining break to advance.
In the later finishing match, a break of 63 was enough for Bingham to secure the opener. Maguire looked in the box seat to level matters in the second but with the reds well spread and the frame at his mercy, the Scotsman missed a simple black off its spot and a grateful Bingham cleared up with a 51 break to establish an early two frame advantage.
Another Maguire mistake in the next left him 52 points behind but a missed Bingham pink to the corner allowed Maguire a vital 67 clearance to the black to pull a frame back. A further straightforward missed red in the fourth allowed Maguire to accumulate a 42 break before laying a telling snooker in behind the green, good enough to draw a foul from Bingham and leave the Scot in position to take the frame with a 50 break and level matters at 2-2.
Bingham restored his lead thanks to an accomplished 68 in the next and it looked for all money like Maguire would level again at 3-3 but with his break on 60, he sportingly called a foul on himself for feathering the white with his cue tip and Bingham cleared superbly to snatch the frame on the black.
Both players squandered golden opportunities to win the next before Maguire won it after Bingham misjudged his positional shot from blue to pink. The Scot needed just one scoring visit to level matters at four frames apiece but Bingham again responded to his latest threat to take the next with a superb 120 clearance, comfortably the highest break of the match.
And Bingham continued where he left off in the previous frame, breaks of 60 and 54 sealing a meeting against Carter.