Victory over Tottenham at the weekend made it three successive Barclays Premier League wins for the first time since May 2011.
The result moved them to within two points of Arsenal, in fifth, and seven adrift of fourth-placed Chelsea, although both clubs have a match in hand.
With games to come against Southampton, Aston Villa, West Ham and Reading - all in the bottom nine - a good run could put Brendan Rodgers' side in an even stronger position by the time former Reds boss Rafael Benitez returns to Anfield with Chelsea on April 21.
"We said it was always going to be tough but we set ourselves the little task of trying to win as many games as we could in the last 10 and that's three in a row now so we've got a chance," Downing told Press Association Sport.
"We've got some winnable games but that's easier said than done. We have to go and win them but from the way we are playing I think we can beat anyone.
"The teams in and around that top half are playing each other so obviously some are going to slip up and drop points and the thing for us is to take each game as it comes and see where we finish."
There is a clear feeling there has been progress made under Rodgers, particularly since January and the form guide backs that up.
After his first 14 matches in charge (up to the end of November) Liverpool had just 16 points and were 12th in the table.
In the subsequent 15 they have collected 29, bettered only by Manchester United and Spurs - and have moved up to sixth.
Downing expects that to continue throughout the remainder of this campaign and, with new recruits arriving in the summer, into the next.
"I think getting everyone back fit, getting our strongest players out there, has helped with everyone in their proper positions," he added.
"I think everything is coming together. There were signs earlier in the season but it was always going to take time to click - now you are seeing some good signs.
"There was going to be a learning process this season, building for the future.
"If we can keep this squad together, add a few more and keep the big players, I think we have a hell of a chance."
A significant contribution has been made by Luis Suarez, who is the league's leading scorer with 21 and who has scored 12 in as many games since December 22.
"If you get players like him who are liable to get a goal out of anything you are always likely to win the game," said Downing.
"That is why he has been talked about as player of the year because every week he is there performing, getting a goal and that is what we need.
"It's not only his goals but he brings other people into play, he is a nightmare to play against with his twisting and turning and he proved that against Tottenham."
Downing himself has enjoyed a rejuvenation in his Liverpool career since the turn of the year - leading captain Steven Gerrard to describe him as a "different player".
The England international, who has scored in his last two matches, said it had helped him playing in a side full of confidence.
"It is certainly a shift playing in my position, getting up and down and trying to nick a goal," said the 28-year-old winger.
"But when the team as a whole plays so well it inspires everyone.
"We look a threat on the attack and we are looking a good team and hopefully we can finish as high as possible with nine games to go."
Injured defender Martin Kelly is hopeful he can make a return from knee surgery before the end of the season.
The 22-year-old has been sidelined since rupturing his anterior cruciate ligament in September but with Jamie Carragher due to retire at the end of the season Kelly has his sights set on the veteran defender's final farewell appearance at Anfield.
"It's a big incentive to be part of that day, whether it's playing or not," he told Liverpoolfc.com.
"I'm at the point now where I've been getting to do little ball drills and stuff without the ball - it's really promising for me at the moment and the medical staff are doing really well with my rehab.
"I'm getting stronger and stronger each day. We've had no setbacks so it's looking really promising."