The 881 was a development of the team's 1987 car, the 871, continuing the Zakspeed philosophy of designing and building both the chassis and engine in-house — a distinction shared on the turbo side in earlier seasons only by Ferrari and Renault among the grid's competitors. The car was driven by Italian veteran Piercarlo Ghinzani and West German rookie Bernd Schneider, who replaced 1987 drivers Martin Brundle and Christian Danner respectively.
The 1988 season saw turbocharged cars facing atmospheric-engined competitors for the last time, but under a restrictive fuel-load cap of 150 litres that negated much of the turbo's power advantage. Even so, the 881 proved unable to outqualify a significant portion of the atmospheric field, a humiliating predicament for a car that in theory had access to more peak power.
Zakspeed's engine was rated at approximately 640 bhp for the season — within roughly 10 bhp of the Honda and Ferrari V6 turbo units and comparable to the Megatron straight-fours used by Arrows. Yet while Honda-powered McLarens won 15 races and took 15 pole positions, the 881 struggled to make the grid at all. Ghinzani failed to qualify on seven occasions. Schneider failed to pre-qualify or qualify ten times. The car weighed approximately 560 kg, around 20 kg heavier than the McLaren MP4/4 and other frontrunning machines, and the excess mass compounded the car's chronic difficulties.
The nadir came at the British Grand Prix at Silverstone, a high-speed circuit that should theoretically have suited turbo machinery. Both Ghinzani and Schneider failed to qualify, making the Zakspeed 881s the only turbocharged cars in the field not to make the grid.
Bernd Schneider's best result of the season came at the German Grand Prix at Hockenheim, his and the team's home race, where he finished 12th. The German rookie had to wait until the Mexican Grand Prix, round four of the season, to qualify for his first Formula One start. At altitude in Mexico City, where thinner air reduced atmospheric engines' power by an estimated 25 percent while affecting turbos by only around 5 percent, Schneider qualified in an encouraging 15th place and ran competitively in the early stages near the edge of the top ten before retiring with engine failure on lap 16. Neither driver scored a championship point across the entire season.
The 881's failure to score points in 1988 had a direct sporting consequence: Zakspeed was required to pre-qualify for the 1989 season, a humbling process reserved for the weakest teams and new entrants. The 881 was the last Formula One car in which the German constructor used its own engine. Its replacement, the Zakspeed 891, was powered by a Yamaha V8 — marking the end of Zakspeed's status as an independent power-unit manufacturer and the beginning of a new phase in the team's efforts to remain competitive in a rapidly changing field.