But Easterby insists that their sixth-place spot is still far from safe as they get set for three derby clashes to finish the season. They play the Dragons at home in-between two clashes with the in-form Blues and the former Ireland flanker is hoping the opportunity to play at the Millennium Stadium can inspire his players in the first of those encounters this Sunday.
"We haven't got that (European) position nailed down yet, especially with three derbies coming up," said Easterby, whose side finished fourth last season before being knocked out by Ulster in the play-offs.
"We will need to be much better in those games against the Blues and Dragons. Hopefully playing at the Millennium Stadium next week will be a big driving force for the players because we need to improve.
"We need to work on how we play without the ball because we coughed up soft tries and didn't take the game away from Zebre when we were in front in the second half."
The Scarlets trailed 13-9 at the break against the league's lowest-placed side but they looked to have put the game to bed when Owens crossed with 27 minutes remaining.
They were unable to finish the Italians off with a third try, however, and Gonzalo Garcia's converted score led to a nervy final 15 minutes in Llanelli.
"We played too much rugby in our own half in the first half and then in the second we started to turn them and gain better field position," added Easterby.
"I thought we worked better in the second half, but we had to overcome a first-half deficit which came from our errors more than Zebre working for the opportunities.
"We scored one excellent counter-attacking try and then one through the forwards. Ken's was a good example of a team try. But we were a bit soft in defence and we allowed them to stay in the game right until the death."