You can safely say that the OH^- from NaOH reacts with the H^+ cations in water.:
There is therefore an excess of HO^- and less H^+.
However, if you know about ionic equilibria and Le Chatelier's Principle this would be the better explanation:
The newly added OH^- react with the H^+, hence shifting the balance to the right and having more OH^- than H^+.
Same thing would happen with NH3, with NH3 reacting with the H^+ and shifting the balance to the right, thus making more OH^- ions and removing H^+ ions.