The victory moved Andretti into second place in the point standings after eight of 17 races, trailing third-place finisher Bobby Rahal, 104-94.
Andretti had been trailing Fittipaldi's Penske-Chevrolet most of the race, leading only one lap, during the first round of scheduled pit stops; but Andretti moved to the front in the first turn on lap 53.
Fittipaldi, trying for his third Cleveland win, immediately slipped to the inside and retook the lead, moving just ahead of Andretti's Lola-Chevrolet. But Andretti darted to the inside on turn three, out-braked Fittipaldi and moved in front for good.
He held off Fittipaldi by 2.25 seconds despite two full-course caution flags in the final 18 laps that bunched the leaders.
The green flag came out for the final time just three laps from the end, but Andretti easily held the top spot, with Fittipaldi more occupied trying to keep Rahal behind him.
Al Unser Jr. barely held off Arie Luyendyk for fourth. Mario Andretti, Michael's father and teammate, wound up sixth, the last competitor on the lead lap.
The battle for third in the standings got tighter, with Luyendyk at 79 points, Unser 78, Fittipaldi 77 and Rick Mears, who went out early after tagging a wall, 76.
John Andretti was running off the pace when he slid off one of the fast turns and slid broadside into a concrete barrier. John Andretti, Michael's cousin, was not injured but brought out the last caution flag on lap 79.
The other yellow came out on lap 68 after CART officials determined that the track was breaking up badly in several spots and took time to clean the surface in the interest of driver safety.
1. (2) Michael Andretti, Nazareth, Pa., Lola-Chevrolet, 85, $78,922, 117.763. 2. (1) Emerson Fittipaldi, Brazil, Penske-Chevrolet, 85, $69,322. 3. (7) Bobby Rahal, Dublin, Ohio, Lola-Chevrolet, 85, $49,972. 4. (3) Al Unser Jr., Albuquerque, N.M., Lola-Chevrolet, 85, $43,422. 5. (8) Arie Luyendyk, Netherlands, Lola-Chevrolet, 85, $39,772. 6. (5) Mario Andretti, Nazareth, Pa., Lola-Chevrolet, 85, $36,122. 7. (13) Scott Brayton, Coldwater, Mich., Lola-Chevrolet, 83, $32,222. 8. (12) Eddie Cheever, Aspen, Colo., Lola-Chevrolet, 83, $28,822. 9. (11) Danny Sullivan, Aspen, Colo., Lola-Alfa Romeo, 83, $25,422. 10. (17) Mike Groff, Northridge, Calif., Lola-Cosworth, 82, $24,742.
11. (20) Buddy Lazier, Vail, Colo., 1990 Lola-Cosworth, 81, $24,402. 12. (19) Didier Theys, Belgium, 1990 Lola-Cosworth, 81, $23,722. 13. (15) Tony Bettenhausen, Indianapolis, 1990 Penske-Chevrolet, 81, $23,722. 14. (21) Hiro Matsushita, Japan, Lola-Cosworth, 77, $16,911. 15. (9) John Andretti, Indianapolis, Lola-Chevrolet, 76, crash, $22,022. 16. (16) Jeff Andretti, Nazareth, Pa., Lola-Cosworth, 76, $20,322. 17. (4) Rick Mears, Bakersfield, Calif., Penske-Chevrolet, 49, crash, $19,472. 18. (14) Ted Prappas, Los Angeles, Lola-Judd, 39, suspension, $14,361. 19. (10) Scott Goodyear, Canada, Lola-Judd, 28, electrical, $19,472. 20. (22) A.J. Foyt, Houston, Lola-Chevrolet, 27, suspension, $19,472.
21. (23) John Jones, Canada, 1988 Penske-Buick, 25, engine failure, $18,622. 22. (18) Willy T. Ribbs, San Jose, Calif., 1990 Lola-Cosworth, 19, oil pressure, $18,622. 23. (6) Scott Pruett, Dublin, Ohio, Truesports-Judd, 4, suspension, $18,622.
French Grand Prix
MAGNY COURS, France -- Nigel Mansell of Britain out-dueled Alain Prost of France over 72 laps for his first Formula One victory of the year.
Mansell's Williams-Renault finished five seconds ahead of Prost's Ferrari at the finish. They were separated by only a few seconds throughout the 192-mile race.
Ayrton Senna of Brazil was third in a McLaren-Honda. Senna, the defending world champion, won the first four races of the season and leads in the drivers standings with 48 points.
Mansell hadn't won since the Portuguese Grand Prix last September. He was leading the Canadian Grand Prix last month but his car broke down on the last lap.
Mansell and Prost were first or second the entire way. The largest margin was 16 seconds for Mansell after a pit stop by Prost on lap 28.