United, who finished fourth in the Scottish Premiership last season, began in convincing fashion at Pittodrie, running out 3-0 victors.
Ryan Dow put the visitors in front in the 29th minute, taking advantage of Ashton Taylor's under-hit backpass to score beyond Jamie Langfield.
That lead was doubled on the stroke of half-time, Gary Mackay-Steven slotting home after the hosts had surrendered possession in midfield.
Substitute Chris Erskine rounded off the scoring in second-half stoppage time to cap an impressive performance from Jackie McNamara's men.
In the day's other game, St Johnstone won 2-1 thanks to goals from Michael O'Halloran and Steve MacLean.
O'Halloran put Tommy Wright's side ahead shortly after the half-hour mark when David Wotherspoon's cross was punched into his path by County goalkeeper Antonio Reguero.
The visitors were two goals to the good shortly after half-time as MacLean got on the end of Scott Brown's cross with a diving header.
County halved the deficit in the 55th minute through Jake Jervis' header, but the hosts were unable to complete the comeback as St Johnstone joined United, Motherwell and Inverness Caledonian Thistle as victors in the opening round of action.
