Owen switched from "us" when Liverpool were winning, to "them" when they lost 3-1.
However, Owen believes Liverpool's lack of European football next season can help their bid for success in the Premier League after a 3-1 defeat against Sevilla in the Europa League final denied them a Champions League spot.
Jurgen Klopp's side finished eighth in the Premier League and missed out on a Europa League spot when Manchester City beat them 3-1 on penalties in the Capital One Cup final.
A Champions League place was on offer to the winners in Basel, but after Daniel Sturridge gave Liverpool a deserved half-time lead over Sevilla, the Reds capitulated in the second half.
Kevin Gameiro equalised within seconds of the restart and Coke took advantage of some slack defending to bag a match-winning brace as Sevilla lifted the trophy for a third successive year.
However, Liverpool's most successful domestic campaign in recent years came in 2013-14, when an absence from European competitions saw them finish second under Brendan Rodgers after they had threatened to win a first league title since 1989-90.
Owen, who scored 118 league goals for the Reds, said on BT Sport Europe: "The final's gone, we haven't won it, we're not in the Champions League.
"This club... you can't rest on your laurels... you can't rest on what's happened.
"It's all looking forward now and I think they've got the right man in place to do the job.
"As he (Klopp) says, time on the training pitch - that's a positive - time to get more belief in the squad and maybe add one or two new faces to it.
"But start looking forward. It's going to be disappointing for a few weeks, maybe even longer, but at some point you've got to put it behind you and start looking to the future.
"Once pre-season starts there will be a new wave of optimism and as I said the last really good season in the Premier League that Liverpool had was when they weren't in any European competition, so maybe use that as an advantage on your competitors."
Owen's former Anfield team-mate Steve McManaman argued that attracting star players to the club was going to be made more difficult by the club's Champions League absence.
"It will be harder to attract the real star players to Liverpool," he said.
"Yes, Liverpool have got an amazing history, they've been in two finals, and that will help.
"The manager will certainly help, his aura will drag players to the team, but the very, very best want to play at the very highest level and that is in the Champions League."