A point away to Bosnia-Herzegovina on Saturday will seal their place in the finals in France next summer.
Robson-Kanu says former manager Speed laid the foundations for the current success before his death in 2011.
"When Gary Speed took over a number of years ago he came in and changed the environment in the football nation," said the Reading striker.
"Before that there wasn't really a desire to win, to be the best, and to do as well as you can.
"But Speedo really implemented that and, when he passed, Chris Coleman came in and has now really taken that on and beyond."
Coleman was appointed Wales manager in January 2012 following the death of Speed.
Wales lost their first five games under Coleman but now the national side need only a point to qualify for next summer's European Championship finals.
"He's really taken the squad and the team forward and given us players the belief and desire to really achieve something," added Robson-Kanu.
"That is what we're on the verge of but, throughout the special moments we've had during the campaign, we've remembered the impact Speedo had when he came in. That will never be forgotten."
Robson-Kanu is expected to retain his place as the most advanced player in Wales' formation to face Bosnia in Zenica.
The 26-year-old has started the past six qualifiers, scoring once, as Wales have remained unbeaten to top the Group B standings.
His work-rate has not gone unnoticed by the Wales fans, who sang his name throughout last month's 0-0 draw with Israel in Cardiff.
"It's some job getting a song for my name and credit to the Barry Horns band," said Robson-Kanu, who has two goals in 28 appearances for Wales.
"It's a massive honour to have them singing my name like that.
"Being a national cult hero is special - but it's handy when you're playing next to Gareth Bale."
Wales striker Hal Robson-Kanu believes Euro 2016 qualification will be the perfect tribute to the late Gary Speed.