The company has an annual turnover of £14 million and the deal ends its 80-year connection with the Chalmers family.
The family are listed at Companies House as the owners of DP&L Group parent company Cortachy Holdings, with Annie Margaret Chalmers the majority shareholder with 50,000 shares.
Outgoing managing director Alasdair Chalmers was listed as the second-biggest shareholder with 26,300 shares.
Mr Bisset, a director and co-founder of Bissets Accountancy & Tax Services, was backed by the firm's senior management team to complete the deal, which went ahead for an undisclosed sum.
He has been working with the board for three years to restructure the group and put a succession plan in place.
Mr Bisset, who is also a director of Air Charter Scotland and Douglas Laing & Co, the Glasgow-whisky bottler, will take over as group managing director.
He said: "We are delighted to be taking control of the group and build upon the good work that has been achieved under the Chalmers' family stewardship.
"I am now looking forward to working with the management team to not only consolidate the current trade but to expanding the group beyond its Dundee heartland."
DP&L, which began life as The Dundee, Perth and London Shipping Company in 1826, employs 50 staff and turned over £14 million in its latest financial year.
It operated passenger services until the outbreak of the Second World War in 1939, and cargo shipping until 1962.
The group underwent significant diversification with Mr Chalmers at the helm.
The outgoing managing director, who is retiring after 30 years, led DP&L out of shipping into travel management, industrial and hygiene supplies, marquee hire and recruitment after succeeding his father in 1987.
The group's divisions include Allison & Stiven (an industrial supplies and marquee hire and manufacture firm), and DP&L Travel, which has bases in Dundee and St Andrews.
It also includes recruitment business Key Personnel (Dundee), which has a strong presence in warehousing and driver sectors.
The Chalmers family acquired the business in 1993 after several changes of ownership.
Mr Chalmers said: "DP&L has prospered by embracing and accepting change. This is a significant development and it ends a family connection of 80 years.
"However, the sale to Alick and the management secures the future of the group. My family and I wish them every success."
Mr Bisset, who stayed with Douglas Laing & Co after the whisky blender and bottler was split into two firms by Fred and Stewart Laing last year, pledged to retain DP&L's customer service ethic.
He added: "While the group's activities are diverse it is our intention to better leverage the strength of each division.
"Over the coming months we will be developing our plans in more detail.
"This will include improving our online presence and increasing business development activity to build upon the group's existing blue chip customer base."