The 22-year-old midfielder scored both goals in a re-run of last season’s final as Spurs again finished second best at Old Trafford.
Gibson, making a rare start, showed marvellous composure with both efforts to put United firmly in control.
Only 16 minutes were on the clock when Anderson cut in from the left flank and rolled a pass invitingly into Gibson’s path 25 yards out. The first-time shot was lethal, flying like an arrow in the top corner and giving Heurelho Gomes no chance.
Gibson’s second was another gem. He played a smart one-two with Danny Welbeck before curling another first-time shot into the opposite corner to seal the win and take United into Wednesday night’s semi-final draw.