There was a frantic start to the match with neither side finding any rhythm as tackles few in from all sides.
Ross McCormack was proving to be a thorn in Hearts' side and his cross into the box was turned wide by Chris Porter before he himself fired a shot wide from 18 yards out.
It wasn't all Well with Marc Fitzpatrick having to clear off the line from a Deividas Cesnauskis shot with his keeper Graeme Smith beaten.
It took a penalty to break the deadlock midway through the first period.
Once again McCormack was heavily involved as his shot was handled by Eggert Jonsson.
McCormack stepped up to take the penalty kick himself and he fired high past the helpless Steve Banks.
The remainder of the first half was a bit of a midfield stalemate, although Well keeper Smith had to look lively to keep out a Michael Stewart overhead kick.
Hearts sent on Christian Nade at the start of the second half and they hit Motherwell with an aerial bombardment as they looked for the crucial second goal in the game.
They should have been back on level terms on 63 minutes but an unbelievable double miss kept Well ahead.
A Jonsson cross was missed by Mark Reynolds and fell kindly for Andrius Velicka and the striker picked his spot but the ball crashed back off the inside of the post.
The rebound fell kindly for Ruben Palazuelos but the Spaniard somehow managed to scoop his shot over from eight yards out with the goal gaping.
As the game moved into the last ten minutes, Nade created a couple of chances for himself but on both occasions the keeper was up to the task of keeping the Frenchman out.
Motherwell managed to hold on and progress to fifth round where they will face Dundee at home.