Robson-Kanu prodded home from close range in the dying seconds of a breathless final 13 minutes that produced four goals.
Dimitar Berbatov appeared to have struck the winner for Fulham moments earlier after driving home Steve Sidwell's pass, but Reading refused to throw in the towel.
The Royals' search for a first Premier League win of the season continues, but they will take comfort from twice hauling in their opponents' lead.
They created the first chance for former Fulham striker Pavel Pogrebnyak, who will have nightmares over the fifth-minute sitter he nodded wide of the left post from five yards out.
The Russian had done the hard work, outmuscling Fulham's defence as he leapt to connect with Nicky Shorey's cross, only to head the ball wide.
Wonderful link-up play masterminded by Hugo Rodallega presented Berbatov with a half-chance, but his hard shot drifted outside the right post.
It was then Rodallega's turn to rue his lack of composure as he sent a wonderful cross by Kieran Richardson straight into the arms of Alex McCarthy.
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Reading surged ahead with a magnificent 27th-minute goal from Mikele Leigertwood who drilled the ball home from 20 yards.
Skipper Jobi McAnuff did the initial damage with an intelligent run that ended with an inch-perfect pass to the unmarked Leigertwood who gave the diving Mark Schwarzer no chance.
Rodallega's wastefulness continued as once again he directed a glorious opportunity straight at the grateful McCarthy.
Heavy rain had given way to sunshine and the capacity crowd were enjoying a lively encounter between two well-matched sides.
Fulham attacked with great speed, but occasionally were left exposed to the counter.
Berbatov yelled in frustration as he sought to offload the ball only to be surrounded by static team-mates, and Reading quickly regained possession.
Jimmy Kebe was finding space down the right with McAnuff just failing to reach one cross.
Moments later Jason Roberts, who again did not wear a Kick It Out T-shirt during the warm-up, darted past left-back John Arne Riise before his deflected shot was bundled to safety.
Reading's ascendancy was brought to an abrupt halt in the 62nd minute when Fulham equalised through Bryan Ruiz.
The Costa Rican forward exchanged passes with Damien Duff before dispatching a tremendous shot struck with vicious force past McCarthy.
Fulham were energised by the goal and began spraying the ball around with greater urgency, spending lengthy spells around Reading's area.
They were rewarded when Chris Baird fired his second goal in two matches, heading home Ruiz's corner in the 78th minute - seven days after condemning Aston Villa to defeat with another late strike.
The score did not come without cost, however, as the Northern Ireland international then limped off having injured himself during the celebration.
Substitute Garath McCleary equalised in the 86th minute from close range as play swung from end to end in a high-octane conclusion that gripped the crowd.
Former Manchester United man Berbatov appeared to have stuck a classy winner, but Robson-Kanu had the final say.