Downing - moved from the wing to full-back for the second half - cut in from the left flank in the 53rd minute to crack home a shot from outside the box.
It secured a 1-0 victory that moved the Reds to the top of Group A, and at the final whistle Rodgers urged the 28-year-old England international, who has struggled to make an impact since joining Liverpool in 2011, to go and soak up the applause of the home crowd.
Rodgers said: "It has been difficult for Stewart since he has been here, and it was just something to help his confidence.
"I think the Kop recognised that it was a wonderful goal at the end and if you can get a round of applause from them, that would mean a lot to anyone - a player, manager, coach, whatever.
"You saw how they reacted accordingly. They gave him a great round of applause and that will hopefully do him good going forward. His commitment was excellent.
"He had to play in a different position in the second half, but he has got the quality to do that."