The support services firm said the deal, which gives it a 10% share of the rail infrastructure market, had been nearly a year in coming.
Chief executive Alan Robertson said: "We have been in talks with them since last November. It was a very important deal for us. We were moving swiftly but Potters Bar disrupted the process slightly."
Responding to fears that the acquisition was risky in the wake of the Potters Bar disaster, which killed seven people and led to concerns over rail maintenance procedures, chairman David Jackson said: "We have gone straight to the top of the industry by buying First, especially in terms of its safety record. We think there are big growth prospects in this business."
The group said the combined turnover of First Engineering and Peterhouse's existing rail business would be £260m. Robertson said the group planned to retain that market share as the market grew. "We are going to have to bring a lot more people into this business," he said.
The group said it would retain the First Engineering name and the management, but was looking for a chief executive to replace Tony Smith, who will become chairman. The firm said it would be looking for external candidates.
The acquisition of First Engineering is part of Peterhouse's plan to reshuffle its business. As part of this, it has put its construction division up for sale. The division includes Totty Construction and Jackson Construction.
Phil Brierley, chairman of Totty, is expected to lead a management buyout of the division next month.
Jackson said he hoped to complete a sale within six months.
He added that the construction division no longer fitted into the group because it was expanding into development.
He said of the operation: "It's a net user of cash rather than a provider. First is replacing a cash-using business with a cash-generating business."
This strategic shift coincided with interim results announced by the group. Its turnover slipped £3.9m to £175.7m in the six months to 30 June, and pre-tax profit was up 10% to £7.04m.