If you measure the total background noise that comes in through the receiver bandwidth, then that's true. We can conceptually measure total noise by connecting a resistor to the IF (the Intermediate Frequency of the spectrum analyser we're using) and seeing how hot it gets, the total thermal power, and this increases with the bandwidth (assuming spectrally flat noise). As we increase the bandwidth 10 times, we get 10x more noise power in.
However, even though the spread signal looks noise-like on a spectrum analyser, as a result of being spread with a noise-like signal, we know something very important about it. We know what the spreading signal is. That means that once we have locked on to the spreading signal, we can average the signal coherently, which means the signal adds as voltage, not as power. What we do is correlate the incoming wideband signal with a locally-generated replica of the spreading signal, to extract the original narrow-band signal.
Now something quite important happens when we correlate and filter the wideband spread signal, because we are also correlating and filtering the wideband background noise as well. If the spreading signal is uncorrelated to the background noise (an assumption we can usually make safely) then this process reduces the power of the background noise by the ratio of the bandwidths. Because our local spreading signal is the same as the original, we retain all of the original signal power, while we reduce the total background noise. And there is your SNR improvement.
A corollary of this is, we need to generate a local replica of the signal to do the demodulation, if it's wrong, then we don't get the SNR improvement. So how do we lock-on in the first place? It doesn't happen by magic. But perhaps that's the answer to another question.