Asante stabbed home from close-range for his eighth goal in 11 league games after superb play by Laurence Maguire.
Joe Piggott had the chance to level the score but Smith, who was adjudged to have fouled Yusifu Ceesay in the box, dived to his right to keep his clean sheet in tact.
Here are Liam Norcliffe’s player ratings fom the game...
1. Grant Smith 9
A second successive clean sheet and a penalty save. Did he even touch Ceesay for the penalty decision? Apparently Altrincham manager Phil Parkinson agreed it was the wrong decision. Two decent stops in the first half and another towards the end. Rowe has given him his chance and he has taken it. Photo: Dean Atkins
2. George Carline 7
Not quite up to his standards against Notts County but still a good outing. After the game Rowe said he switched Carline to left wing-back at the start of the second half because they could not get him into the match as much as they would like. Photo: Dean Atkins
3. Will Evans 7
It looked like Evans was going to be in for a long night when the lively Ceesay turned him inside out in the opening stages. But credit to the skipper for getting to grips with him by being aggressive and first to the ball. Produced an outstanding last-ditch challenge to deny Kosylo who was through on goal in the first half. Photo: Dean Atkins
4. Haydn Hollis 7
Altrincham's pace, movement and trickery caused him some problems in the first half. His error led to a chance for Kosylo before Evans got him out of jail. But overall he did well and it is another clean sheet. Photo: Dean Atkins