You can do this with the right FET acting as low side switch:
With 4.5 V gate drive, this FET goes down to 45 mΩ. With your 1.6 A load current, it will drop 72 mV and dissipate 115 mW. That's fine since its absolute maximum rating is 800 mW at 70 °C ambient.
In this case the gate can be driven directly from a 5 V logic digital output.
If it matters that the LEDs don't come on for the few 10s of ms while the microcontroller is starting up and before it can set the digital output to low impedance and drive it low, then add a small pulldown on the gate to ground. 100 kΩ will do.
Logic Level FETs
I see there is some discussion about other FETs and what is really needed in this application. Lots of very beefy FETs are available if you can give them 10 to 12 V of gate drive. While that could be done here by using a gate driver chip powered from the 12 V, it is unnecessary.
There are FETs that can get to nicely low Rdson with less than 5 V gate drive. The tradeoff is that they can't withstand a lot of D-S voltage, usually up to 20 or 25 V, sometimes as high as 30 V. That's no problem in this case since only a 12 V power supply is being used.
The marking term for these are "logic level", because digital logic used to be 0-5 V signals. The datasheets promise what the maximum Rdsonwill be with a little less than 5 V, like 4.5 V. That allows you to safely drive them from 5 V CMOS logic outputs.
Of course nowadays, logic isn't 5 V anymore, so the "logic level" marketing term isn't very useful. You have to dig around to see which of those FETs are also specified for a little less than 3.3 V gate drive. The IRLML2502 I show above is one of these, although that isn't relevant in this particular case since the logic signals are actually 5 V.
However, you should not use a FET with 5 V gate drive unless the datasheet tells you what you get. The IRF530 mentioned in another answer is one of these. Here is the relevant snippet from its datasheet:
Note that a maximum of 90 mΩ Rdson is promised, but only at 10 V gate drive. Nowhere in the datasheet does it tell you what Rdson you get at any other gate voltage. All it mentions is that the gate threshold voltage is somewhere from 2.0 to 4.0 V. But, that's where the FET starts to come on, which is also defined in the datasheet as allowing 250 µA of drain current. That's a totally irrelevant spec when you want to switch 1.6 A.
The IRF530 is gross overkill in other parameters. That by itself does no harm, but it's a totally inappropriate FET for this case. All you know is that with over 4 V of gate drive, you get more than 250 µA of drain current.
Wishful thinking and designing to specs not promised in the datasheet have no place in engineering. The fact that a part is included in some hobbyist kits for use in a certain unspecified way is no evidence of anything other than irressponsible engineering. Individual parts found to work in certain ways is also not relevant. There is nothing guaranteeing that the next part out of the box will exhibit that same behavior.
