Southampton made sure there was no repeat of their weekend collapse against Wolverhampton Wanderers as they claimed an impressive 2-0 win at Crystal Palace on Tuesday.
Redmond smashed home in the 22nd minute when he was played in behind the Palace defence by Jack Stephens.
Southampton doubled their lead when Armstrong whipped in a curling effort after 48 minutes.
Palace did improve but Southampton were solid as they claimed a fourth successive away league win.
Southampton have now won five of their last seven league games and have moved from the relegation zone after 17 matches played to the top half of the table after 24, just three points behind fifth-placed Manchester United.
"I think it was one of our best performances of the season," Hasenhuttl said. "It's not so easy to win here.
"In the second half it was important that we did not make the same mistakes as we did against Wolves, but we were very clinical and came up with a beautiful goal from Armstrong."
Palace produced a remarkable display to draw away at Manchester City on Saturday, but Roy Hodgson's side looked heavy-legged and struggled to deal with Southampton's pace.
With no FA Cup tie at the weekend they will not be in action again until Feb. 1 and Hodgson hopes his players can recharge.
"I think it was a game too far for us," he said. "We have been battling with the same group of players.
"We have had tough games, battled through brilliantly, but that all caught up with us today and we deserved to lose the game because Southampton were the better team."
After Redmond's superb finish Saints should have added to their lead before the interval with Palace keeper Guaita making several interventions.
Armstrong did finally reward Southampton's dominance and this time there was to be no collapse by the south coasters.