Both teams won their last 16 ties in midweek, Rose winning 4-0 away to Dundonald Bluebell thanks to goals from Lee Currie, Neil Martinyuik, Kieran Hall and George Hunter.
Bo’ness saw off Petershill 4-1 in Wednesday night’s last 16 match at Newtown Park thanks to goals from Nick Locke, Ryan Stevenson, Gregor Nicol and Lewis Hawkins before Petershill netted a late consolation on the counter-attack.
“It was a very convincing win and I was pleased with the way we passed the ball at times,” said Bo’ness boss Max Christie.
"Bonnyrigg away in the next round is a tough one. We’ve gone down there this season, competed well and got beaten (a 3-0 loss on February 19).
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"And, in the last two games we’ve played against them, we gave away a total of six penalties. I think we need to be wary of that when we go back there.
"Looking back, I think four of them were penalties and you could argue against the other two. The last game we played there was 0-0 with 20 minutes to go and we were just giving them gifts.
"So we’ve got to go down there, play like adults, stand up and make sure there are no crazy penalties given away.
"It’s down to the players’ decision making and how good a player they are, I suppose.”
Following this Saturday’s Lowland League Cup trip to Spartans, Bo’ness have a home East of Scotland Cup quarter-final against Stirling University on Tuesday night.
Victory in that game would set up a semi-final against the winners of Linlithgow Rose and Hill of Beath, who meet on April 9.
Next up for Bonnyrigg is a trip to Rangers B in the Lowland League on Tuesday night.