Sharma compiled a round of 70 and finished on four under par, eight strokes behind runaway leader Brian Harman.
He downed a monster eagle putt at the fifth then reeled off 10 successive pars before his only dropped shot of the day.
Jyoti Randhawa holds the record for the best result by an Indian at the Open, finishing tied for 27th at Troon in 2004.
Sharma played in all four majors in 2018, but only survived the cut at the Open, finishing 51st.
Barring a calamitous final round on Sunday he looks primed to better that and surpass Randhawa's feat.
"Very pleased with myself. I played solid. Just one shot off the top five," he said.
"Played great all three days. Scrambled really well. Made a lot of crucial putts, which I wasn't holing in the past month or so."
Sharma, who learned how to play on Army courses in India courtesy of his father who was in the military, said he would just carry on grinding it out on Sunday.
"I'll stick to my processes that I've done in the first three days, maybe make a few adjustments on things that I need to do, just go in with the same mentality that I have had for the first few days," he said.
"Tomorrow is a new day. I don't know when the Open will be back here, so fourth round at a major, Liverpool, I'm really happy, and I'll give it my best."
The best finish by an Indian player at a men's major is Anirban Lahiri's tied fifth finish at the PGA Championship at Whistling Straits in 2015.
