The Addicks dispatched Bolton 1-0 to secure their place in the fourth round of the Carling Cup and give Dowie some breathing space.
Charlton are bottom of the Premiership and reports said the former Crystal Palace manager had just two games to save his job.
Reed said: “Iain is very happy and has been positive all the way through this. We’re not letting the pressure affect us.
“This is a club where people are patient.
“The fans cheered us off tonight and we’ll move on.
“Iain has never been off track. It’s been frustrating but he’s been positive all the way through it.”
Marcus Bent scored the game’s only goal, rifling home Jimmy Floyd Hasselbaink’s pinpoint cross in the 17th minute, stretching Charlton’s unbeaten run to two games following Saturday’s 0-0 draw with Watford.
Winger Jerome Thomas was stretchered off in the first half after being caught with a knee from Stelios but Reed hopes the problem is nothing more than heavy bruising.
He said: “Jerome is being assessed at the moment. He took a clip on the ankle and then a knee at the top of the hip. Hopefully it is bruising.”
Bolton boss Sam Allardyce felt his side should have been celebrating their passage into the fourth round and blamed sloppy finishing for the defeat.
“We should have won tonight, there’s no question about that. We created enough clear cut opportunities in the second half to finish them,” he said.
CHARLTON: Myhre, Young, El Karkouri (Sankofa 82), Fortune, Hreidarsson, Hughes, Holland, Kishishev (Rommedahl 88), Thomas (Ambrose 40), Hasselbaink, Marcus Bent.
BOLTON: Walker, Hunt, Meite, Ben Haim, Teimourian (Smith 56), Giannakopoulos, Speed (Campo 63), Tal, Vaz Te, Davies, Pedersen (Fojut 73).