The Perth men secured another three points, following their win over Hibernian on Saturday, with a 1-0 victory over St Mirren on Tuesday night.
Steven MacLean scored the winner for St Johnstone as Davidson again stood in for boss Tommy Wright on the touchline, although the Northern Irishman has been released from hospital after gallbladder surgery.
Wright's side grabbed notable away wins over Rosenborg and FC Minsk in this year's Europa League qualifiers but trail fourth-placed Dundee United by six points in the race for this year's European spots.
However, Saints' assistant boss Davidson said: "Everything is a possibility. I know that is a cliche but it is.
"We have two massive games coming up - that's how we are looking at it. We're not sitting here thinking it is job done because we have got into the top six.
"We're now thinking about where can we get to. If we can win as many games as possible then we can put pressure on Inverness and Dundee United."
St Johnstone rested 22-goal top-scorer Stevie May but still had enough to see off sorry St Mirren.
"Stevie was just rested tonight," said Davidson. "He has been running on empty for the last four or five games.
"But, after we clinched the top-six place, it gave us our chance to rest him.
"He needs that rest. He's been playing for Scotland U21s as well but he will be back buzzing for Saturday now.
"But we showed tonight we have other options. Michael O'Halloran came in and gave us a dynamic threat up front with his pace.
"He worked well with David Wotherspoon and Lee Croft, so we do have different threats available to us when Stevie isn't playing."
The defeat is second-bottom St Mirren's seventh in their last eight league games and leaves them two-points adrift of Partick Thistle in the battle to avoid the relegation play-off spot.
"I am flat," admitted dejected boss Danny Lennon. "Especially because of what is at stake.
"It was a very disappointing performance. The players know what is at stake, our livelihoods in the Scottish Premiership.
"But we have to make sure this club remains in this division. There have been too many bricks laid in the foundations here for that to fizzle out."