Hosts Hibs created plenty of chances but were unable to convert any of them and looked set for frustration until Wotherspoon struck 11 minutes from time to set them up nicely for their date with Celtic at Hampden Park next weekend.
Easter Road passed a late pitch inspection after heavy rain and Leigh Griffiths proved to be the chief threat for the hosts, putting a low volley wide of the target early on.
Alex Harris also looked lively for Hibernian and he won a corner which Griffiths sent over and Jordon Forster headed over the crossbar.
Dundee goalkeeper Steven Simonsen then pulled off a fine save to deny Griffiths and the striker's follow-up shot seemed to be blocked by a Dundee arm, but the referee waved play on.
Harris was next to try his luck for the dominant hosts but he shot straight at Simonsen after making room for himself on the edge of the box.
Griffiths leads charge
The second-half began in similar vein to the first, with the hosts charging forward, led by Griffiths.
The on-loan Wolves man sent over a lovely cross from the left but it flashed across the face of goal, with Ross Caldwell not quite able to connect.
Dundee then began to pose a threat and Forster produced a key clearance, before Lewis Toshney shot over when quite well-placed.
But Hibs were soon in charge once again and Griffiths volleyed just wide after latching onto a great pass from Paul Hanlon, before Harris went on a mazy run and then teed up Lewis Stevenson, who blazed high and wide from close range.
Dundee were eager to end on a high themselves, however, and Ryan Conroy was unlucky to see a superb low volley go just wide of the post after he had got on the end of a Gary Harkins cross.
But it was Hibernian who got the breakthrough, even though both Griffiths and Harris had been substituted.
Stevenson crossed from the left and, just a few minutes after coming on, Wotherspoon was on hand to find the target with a powerful header from 16 yards out which secured maximum points for Hibs.
Griffiths therefore finished a winner on his final start at Easter Road before he returns to Wolves after almost two years on loan.
And beaten Dundee may yet get a reprieve from the drop because the SPL board are meeting on Monday to decide if Hearts should receive an 18 point deduction because of the insolvency of their majority owners UBIG.