Need a level shifter, is there an IC which does what I want?
Sure, there are myriads of level-shifter ICs.  Here is one that meets your needs and is fairly cheap.  There is only one unidirectional channel so you'd need three for your example.

This one will do four channels in one IC and is a little cheaper per channel.

Since these parts may be obsolete someday and to keep this answer (ahem) "timeless," here is the procedure to find this in general:
Go to DigiKey (or any vendor, but I like DigiKey's filtering), search "level shifter" and then start filtering by "unidirectional" and where the Vin supply (2V) falls between the range, and where the Vout supply (5V) falls between.

One last thought, I wouldn't use the circuit that you've breadboarded because there is no real path for a logic LOW.  It may be working for you, but will depend on the load your driving having internal losses that eventually pull the load low.  In other words, it should be a floating output but it goes low, meaning high impedance and noise sensitive.  Use a level-shifter IC like I've pointed you to here and actually drive the signal low to avoid other problems down the line due to noise.