With his tap in in the second half - a goal which ultimately proved decisive - Ziyech became the joint-highest scoring player from his country in Champions League history, alongside former Arsenal star Marouane Chamakh
It was a much-needed goal for the 28-year-old in what was a useful victory for Chelsea, who struggled to break down Malmo in the first half
The visitors had a penalty shout early when Kai Havertz was brought down in the box, but the referee rightly didn't give it with replays showing the German had kicked the defender's foot.
Ruben Loftus-Cheek glanced a looping header over the bar following a good delivery from Ziyech ten minutes later.
The 25-year-old, who has been reinvigorated under Thomas Tuchel this term, curled an effort just over the bar moments later as the visitors' pressure mounted.
Ziyech fizzed a shot towards goal following a delightful piece of skill, but his effort was easily gathered by the Malmo goalkeeper.
Havertz should have opened the scoring on the half hour mark after being played clean through and failing to dink over the keeper.
Antonio Colak nearly gave the hosts a shock lead seconds before half-time as his volley snuck just to the right of the post and the sides when into the interval all square.
The impressive Loftus-Cheek once again tested Malmo's resolve after leaving two defenders for dead before striking a shot straight at the goalkeeper.
The European champions finally broke the deadlock when Ziyech tapped home Hudson-Odoi's superb delivery from out wide.
It was a much-needed goal for the Morocco international, who has struggled to establish himself as a first team regular since his £33million move from Ajax last summer.
And on talkSPORT commentary duty, David Connolly couldn't hide his admiration for the delivery of Hudson-Odoi.
"This is put on a plate by Callum Hudson-Odoi, lovely little one-two between him and Kai Havertz," he told talkSPORT 2.
"He gets his head up early and could have carried on, but spotted Ziyech at the back post and this is a beautiful cross.
"It is bent away from the goalkeeper, so he can't come for it. The defenders can't get it either. Ziyech had a simple tap in, brilliant wing play and they didn't do that enough in the first half."
Thiago Silva and Havertz once again came close for the Blues, with a goal-line clearance and Malmo goalkeeper Jonas Dahlin keeping out a second for the visitors.
Christian Pulisic, making his return for Chelsea in the final few minutes after a long injury lay-off, spurned a simple chance to get a goal but ultimately Thomas Tuchel's men had enough to get the job done.