It was Kane's first England goal since 2019 while Mount has now scored or assisted in three consecutive games for his country.
Albania initially made things difficult for England but once skipper Kane headed home in the 38th minute for his 33rd goal for his country it was a relatively straightforward assignment for Gareth Southgate's team.
Kane also hit the woodwork, as did Phil Foden, before midfielder Mount doubled England's lead just past the hour mark with a deft finish from Kane's pass.
Albania were left to rue an early chance squandered by Myrto Uzuni, after which they rarely threatened an England side who followed up their 5-0 rout of San Marino on Thursday with a sure-footed display against much stiffer opposition.
They top the standings with six points from two games ahead of their home clash with most-likely group rivals Poland at Wembley on Wednesday. Albania have three points following their victory over Andorra in midweek.
Kane did not hit the target in six appearances for England in 2020 and was deprived of closing the gap on record scorer Wayne Rooney when he was rested for the San Marino game.
But the Tottenham forward got his total ticking again when he escaped his marker to glance in Luke Shaw's cross to settle his side down after a disjointed start.
"I thought it was a good performance, it's never easy playing away against a tough side," Kane told ITV.
"Maybe we were a bit slow to start but we changed the formation a bit halfway through the first half and got more runners in the box.
"Getting the goal before halftime was great and we controlled the second half well."
England had struggled for fluency in the early stages on a clingy pitch and Albania had the first real chance when they seized on a loose pass and broke forward dangerously but Uzuni blazed over to the relief of the visitors.
Kane should have made it 2-0 when he met Raheem Sterling's dangerous cross at the far-post but his half-volley crashed against the face of the crossbar.
England were far sharper in the second half and Foden, part of a three-pronged England attack, forced a fine save from home keeper Etrit Berisha who turned his shot against the post.
Albania gifted away possession in a dangerous area in the 63rd minute and were duly punished as Kane slipped a pass to Mount who calmly dispatched his shot past Berisha.
"I thought his performance was excellent," Southgate said of Mount after the match.
"He is an exceptional player, he finds space intelligently, he manipulates the ball very well, he creates chances and he scores goals.
The only blemishes for England were yellow cards shown to Kyle Walker and Kane.