The winger, operating in a deeper role in the second half, cut inside onto his weaker right foot before rifling past Vladimir Gabulov on the edge of the box.
His manager pushed him into the centre of the pitch to accept the praise and believes it will have been a huge confidence booster for him.
Rodgers told the club's official website: "It was a wonderful strike by Stewart and he scored a great goal. It's been difficult for him since he's been here.
"I think [telling him to approach the Kop] was just something to help his confidence. I think the Kop recognised it was a wonderful bend on the strike and if he can get a round of applause from them, then that would mean a lot to anyone; to a player, manager or a coach.
"And you saw how they reacted accordingly - they gave him a great round of applause and that will hopefully do him well going forward. His commitment tonight was excellent. He had to play in a different position in the second half but he's got the qualities to do that. It was a good night for us."
|6/1||Liverpool are 6/1 with PaddyPower to secure a top-four finish in the Premier League
Rodgers was particularly impressed with his side's third consecutive clean sheet and with the way the Reds pushed forward throughout.
He continued: "Anzhi have got a terrific group of players and so to get a win and keep another clean sheet was excellent. But I thought we deserved it, I thought our intent in the game was to go forward and in the second half we stepped it up and did very well.
"There's been a massive investment there as a team and a club, both in terms of the playing staff and the management.
"I've been an admirer of Samuel Eto'o for years - he's a top, top striker and there's a number of other players in the side [who are dangerous]. So we knew it was going to be a tough game.
"[In terms of the away leg], I will do what I have done in every game since I've been here and when we travel abroad we will give great respect to the opposition and field a team that can win the game.
"But it was very much about the squad tonight. I always look to pick a team to win the game. It was our third clean sheet and we looked like we could score tonight. But the most important thing was the victory, which was very good."
Up next for Liverpool is the Merseyside derby and Rodgers insists his squad will be in peak condition despite a number of players out with various injuries.
He added: "For us it's just about taking it one game at a time. The squad is not overly big but the condition of the players is very good so I have no concerns about that.
"Their work rate in training and in the games has been very good. So we'll recover well. It was a very good performance and result tonight and now we are all really looking forward to Sunday."