The champions dominated from start to finish to earn their fifth league win in a row, with Argentine striker Martinez getting the scoring up and running with a bullet header in the 29th minute.
Martinez could have doubled his side's advantage before halftime, but saw a tame penalty saved. The hosts did not have to wait long for their second, however, as Alexis Sanchez, on a rare start, volleyed home five minutes after the break.
Cagliari were dead and buried from that moment on, and Hakan Calhanoglu soon added a third.
Martinez completed the scoring in the 68th minute with a delicate finish after bringing Nicolo Barella's inch-perfect long pass under control, while a host of further chances came and went to make the win even more comprehensive.
With AC Milan and Napoli both dropping points this weekend, Inter took full advantage to hit the summit, one point ahead of city rivals Milan, while Cagliari dropped to 19th having taken 10 points from 17 matches.
Milan and Napoli's long unbeaten start to the Serie A season meant champions Inter, under new coach Simone Inzaghi, have been playing catchup for much of the campaign.
But as their Scudetto rivals have faltered, Inter have hit top form, just ahead of the Serie A winter break, with Cagliari no match for the in-form champions.
Visiting goalkeeper Alessio Cragno was kept busy all match, with Inter having 15 shots on target in the 90 minutes, and it was his brilliance that managed to keep the hosts at bay for 29 minutes.
Inter's 14th goal from a set piece this season, however, broke the resistance, with Martinez powering a brilliant header home from Calhanoglu's corner.
After Denzel Dumfries was upended by Cragno just before the interval, Martinez looked set to double his tally, but the overworked Cagliari stopper made a fine save.
Sanchez was enjoying his rare run out and took his first Serie A goal of the season expertly, volleying into the bottom corner.
The Chilean forward then turned provider for Calhanoglu, who unleashed a stunning strike into the top corner from 25 metres out, before Martinez netted his second and 10th Serie A goal of the season to put the icing on the cake.