I'm not sure what you mean when you say, "(without control cable)". For starters, does the fan you bought have a common 4-pin connector on the end of its wires, exactly like those on most case ventilation fans? If it does, those should ALWAYS fit onto any standard mobo 3-pin fan port connector. And no, there are NOT "lots of ways for it to work". There is only one way it fits. The mobo male connector had a plastic tongue sticking up by the 3 pins, and the fan's connector has a slot on its side so you can only plug it in one way. Any standard 4-pin fan CAN work properly when connected to a standard 3-pin fan port. That includes the ability of the mobo's automatic fan speed control to manage that fan's speed.
Now, IF that is the connector type on your new fan, and IF your graphics card has a standard 3-pin fan port on it, the same should apply: you should be able to plug the new fan into the graphics card's connector and it will work under automatic control.
The fact that you say it cannot work makes me think that there is something odd about the connectors here. Look carefully at: (a) the connector on the end of the wires from your OLD graphics card's fan: (b) the 4-pin connector on the new fan; and, (c) the 3-pin fan ports on your mobo. They should all look to be about the same size and should fit together. The only difference should be that the new 4-pin connector is just a bit wider with an extra hole. If that is NOT what you see, tell us what actually is there. Is one type of connector larger, or a different shape? In each case, how many wires come from the fan motor to its connector? What is the "control cable" you mentioned?