On paper the British number three's 6-3 6-2 7-6 (2) victory was an upset but, given Evans' form and prowess on grass and his opponent's lack of form, it would have been a disappointment if he had not found a way through.
Things got a little tricky in the third set after Evans had first held match points at 5-2 but he got over the line at the fifth time of asking in the tie-break and then struggled to hold back tears.
Evans last reached round three in 2016, losing to Roger Federer, who has become a semi-regular practice partner, with the pair hitting together at the All England Club on Wednesday.
The Brummie is playing at least as well as he was before the self-inflicted career sabotage that took him away from the top level for the best part of two years.
What is clear is how much the 29-year-old wants to make the most of the rest of his playing days after finally getting to a place where he is happy to dedicate his life to tennis.
This was almost an all-British clash, with James Ward leading Basilashvili by two sets to love in the first round only to lose out 8-6 in the fifth set.
Here the Georgian started strongly, pounding ground strokes, and Evans did well to stay with him, saving a break point in the fourth after they had exchanged breaks in the opening two games.
Evans was unable to play his attacking game for the most part but what he did superbly was not to give Basilashvili any free points and that hard work paid off with two breaks of serve to take the opening set.
Another break to start the second set put Evans fully in control of the match and he began to toy with Basilashvili, dragging him into cat-and-mouse rallies and drawing him forward before dinking the ball at his opponent's feet.
In a repeat of the second set, Evans moved 3-0 ahead in the third and held two match points on Basilashvili's serve at 2-5.
It seemed merely to be delaying the inevitable when Basilashvili saved both but the 18th seed threw caution to the wind, charging to the net at every opportunity, and saved a third match point before breaking back.
A fourth then came and went as Evans was pegged back to 5-5 but in the tie-break Evans got back on top once more and he moves through to a clash with either 13th seed and former finalist Marin Cilic or Portugal's Joao Sousa in the last 32.
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