A surge in form has seen Sam Allardyce's team produce a four-match winning run that has lifted them out of the bottom three and into 10th place, seven points clear of danger.
West Ham face an Everton side on Saturday who have lost three of their last four matches, but the winger is refusing to underestimate them.
"We're not out of the woods yet because we've still got games to win, starting at Everton," Downing told the club's website.
"Everton is always a difficult place to go, even when they're not playing that well.
"The crowd are really on top of you, (manager) Roberto Martinez is doing a good job and his team is playing really well using a different way of playing to the past.
"We've been a bit lucky at times recently, but in the last couple of weeks it's all come together. We've been due some luck.
"Everyone is back now from injury and suspension and that's what we've been
Andy Carroll is available after his three-match suspension but Downing feels it will be a tough decision for Allardyce to pick him ahead of in-form striker Carlton Cole.
"It's a difficult situation for the manager, but one he will probably like because he's not had a decision like this to make for six or seven months!" he said.
"It's obviously a big decision because Carlton has come in and done really well and earned himself a contract with his goals, while Andy is back to get some goals.
"Whoever starts is a good option and it makes it easy for us wingers because they are big target men who we can play off.
"It's a difficult decision for the manager, but a good one to have."