For what it's worth, I'd take the Sako, although I'm a big fan of the Anschutz rimfire rifles. The newer Anschutz CF rifles, like the 1771, have the bolt lock up in an insert installed in the barrel. Perhaps a manufacturing compromise. However, it pretty well rules out rebarreling or truing the action the way you could with a standard action in which the locking-lug recesses are in the action. This manufacturing detail is not unique to the new Anschutz CFs, but is, at least in my opinion, less desirable than the more usual design with the bolt locking up in the action.
Another negative for me of the 1771 is aesthetic. The removable magazine extends about 1/2" below the belly of the stock, and not only looks pretty awful (at least to my eye), but also would seem to get in the way of a comfortable hold on the rifle while carrying it in the hand.
Finally, although the Anschutz 5097D single-stage trigger on the 1771 is a very-decent trigger (adjustable down to about 2.25 lbs.), you can order the Sako M85 Varmint model with the really-nice set trigger option. This trigger will function in non-set mode over about the same weight-of-pull range as the Anschutz, but the set feature--allowing a let-off of a few ounces--is really a great option, and would, I believe, be right at home on a varmint rifle.