A sumptuous first-half half-volley from playmaker Lee Currie, who had earlier seen an effort strike the outside of the post, had the hosts ahead midway through the opening 45 before George Hunter's deflected strike doubled Rose's advantage seven minutes after the restart following determined play from wing-back Dean Brett.
Teenage defender Bosun Lawal, who joined Celtic from Watford on a three-year-deal last week, was outstanding throughout and he reduced the arrears with a neat finish from a free-kick just before the hour mark.
The full-time visitors upped the tempo in search of an equaliser but the Rose defence stood firm to secure a precious opening-day win.
Rose manager Robbie Horn said: "I'm delighted with the team's performance. It was hard work out there and obviously the heat played a big factor in it as well. They are a good side, fit boys, they move the ball well and try to play the right way and, to be fair to them, they stood up to the physical challenge.
"Over the piece I think we just shaded it and I've no doubt they will take points off teams so credit to them, they stuck to it and put us under some pressure. They are taught to play the right way but so do we.
"We had to mix it up and be physical at times and try and intimidate them as well. Keiran [McGachie] worked his socks off and really put in a shift and that made a huge difference getting in and about their back-line. He was winning flick-ons, holding the ball up and got us up the park.
"It was a bit of the unknown before the game so there was a bit of nervousness. We didn't get their squad list until 10.30pm the night before the match so we really didn't know who we would be facing. I'm thrilled to get the three points."
Celtic B boss Tommy McIntyre said: "It was a really tough game but we expected it. Bonnyrigg are a really physical team. It's going to be a learning curve for the players, they are going to be uncomfortable and I thought at times they were, which is good, because it takes them out that comfort zone. There were a few good performances and with any luck we might have got a draw."
Meanwhile, Edinburgh University fought back twice to earn a 2-2 draw with Broomhill at East Peffermill.
The visitors had the upper hand in the early stages and took the lead in the 15th minute through Andy Monk's off-the-post finish.
The students equalised just before the half-hour mark as full-back Max Condie got on the end of Matty Brown's cross. However, the Glasgow-based side restored their advantage just five minutes later through another Monk finish.
Dorian Ogunro's men were rewarded for their efforts in the final minute through Ben Spellman's wonderful strike that flew into the top corner from 25 yards that salvaged a point for the uni.
Elsewhere, Civil Service Strollers went down 1-0 at Dalbeattie Star, Lewis Sloan's 12th-minute penalty at Islecroft proving the difference between the sides.