The Kop boss has come under scrutiny after a difficult season which was magnified when the Reds lost to Aston Villa in FA Cup semi-final on the weekend.
The Reds, who sit fifth on the Premier League table, also bowed out of the Champions League in the group stage this season and Rodgers recently admitted that qualifying for Europe's top flight for next season is unlikely.
Of the apparent failures on Merseyside this term, Owen told BT Sport's 'Football Tonight'.
"He is under pressure no question about it," Owen said. "I think he should be given more time, I really do.
"It was only 12 months ago when, if you go to Anfield every week, last season was the best football and the most exciting time to be a Liverpool fan in years.
"I know they won the Champions League and I know we won trophies before that, but that was the best team you know, the best time to be a Liverpool fan.
"One slip away from winning the Premier League and I think Brendan Rodgers is a very good manager and certainly deserves another year just based on that season.
"It has been a disappointing season you know, I'm not saying it isn't but I do think there are so many positives as well, I don't think they are in a worse position than when he took over so I feel he does deserve another year."