Huddersfield Town 3 Grimsby Town 0
Former Mariners striker Jevons, on loan from Bristol City, persuaded City boss Gary Johnson to let him play in the FA Cup.
And Huddersfield were rewarded when Jevons' double secured a place in the third round, before sub Luke Beckett made it 3-0 late on.
Grimsby, facing a fight to stay in the Football League, played well in the first half but were grateful to second-choice keeper Gary Montgomery for three important saves.
The lively Malvin Kamara was the dangerman throughout and Montgomery twice denied him.
Kamara's first-time snap shot was kept out by Montgomery's reflex stop and he then beat away another clean hit.
Defender Frank Sinclair played a neat one-two and drilled a low, hard shot at goal but Montgomery held firm.
Huddersfield cut loose after the break and Jevons broke the deadlock eight minutes after the restart.
An Andy Holdsworth corner caused confusion in the Grimsby defence and, in a goalmouth scramble, Jevons stole in at the back post to fire into the roof of the net.
Jevons made it two in the 62nd minute when Kamara went clear down the right and centred. Jevons, unmarked, fired home.
The home side then piled on the pressure but Jevons took one touch too many as he bore down on goal and was denied a hat-trick chance in the 70th minute.
Jevons, who is lacking match fitness, was replaced to a standing ovation by Danny Cadamarteri 16 minutes from time.
Beckett came on for Andy Booth and the two subs combined to wrap up victory.
With five minutes left Cadamarteri's twisting and turning earned him space but he rasped his drive against the post. The rebound fell for Beckett, who tapped in his sixth goal of the season.