The hosts took the lead in just the fifth minute when Shuhei Akasaki opened the scoring at the Kashima Soccer Stadium, but the sides were level at the break thanks to an own goal from Takeshi Aoki.
Shoma Doi restored Kashima's lead in 65th minute, only for Patric's eighth goal of the campaign to once more level proceedings seven minutes later.
It looked as though that is how the match would remain until Lins struck in the third minute of injury time to extend Osaka's winning league run to six matches.
The result sees Kenta Hasegawa's men leapfrog Kashima into second due to a superior goal difference, with the two teams seven points adrift of leaders Urawa Reds.
Urawa were seeking a return to winning ways after the surprise loss at strugglers Cerezo Osaka last weekend and they had to come from behind to defeat rock-bottom Tokushima Vortis 2-1 at the Saitama Stadium.
Yu Eto put the visitors into a shock 34th-minute lead, only for Yosuke Kashiwagi to restore parity four minutes before the break.
The expected turnaround was completed shortly after the hour mark through Daisuke Nasu, a goal that severely hinders Tokushima's unlikely hopes of avoiding the drop.
Kawasaki Frontale's hopes of reaching the AFC Champions League spots were dealt a blow as Leo Silva's double helped Albirex Niigata to a 3-0 win, but Sagan Tosu's own chances were boosted by a 1-0 triumph over Yokohama F Marinos.
Cerezo Osaka were unable to build on the Urawa Reds victory as they were comfortably beaten 3-0 by fellow relegation-threatened side Shimizu S-Pulse, while Omiya Ardija picked up a vital three points in their 1-0 win at fellow strugglers Ventforet Kofu.
Vegalta Sendai remain mired in relegation trouble, but picked up a timely three points in a 1-0 success at home to Tokyo, Shingo Akamine scoring the only goal with 15 minutes remaining.
In other matches, Nagoya Grampus were 3-1 victors at Vissel Kobe, while Kashiwa Reysol were held 0-0 at home by Sanfrecce Hiroshima.
