Lee Hughes' 20th goal of the season and a superb 25-yard free-kick from Ben Davies had County dreaming of a remarkable upset in front of their own fans at Meadow Lane.
The Latics, though, had other ideas as first Jason Scotland finally ended his goal drought since a £2million summer move from Swansea, pulling one back in the 53rd minute of his 21st appearance for the club.
Thirty minutes later, and following three fine saves from debutant goalkeeper Vladimir Stojkovic, 70th-minute substitute Ben Watson headed home the equaliser to set up a February 2 replay at the DW Stadium.
Martinez described their first-half performance as unacceptable, and was relieved to have come away with another shot at reaching the fifth round.
"I want to give credit to Notts County because the way they started the game, they were outstanding and deserve the replay," reflected Martinez.
"Many of my players have never played in the FA Cup, and I don't think they expected what they experienced in the first half.
"That's not an excuse because we were slow, sloppy and encouraged pressure from Notts County, and we had no direction in our play.
"We gave them a well-deserved two-goal lead, and for us it was a real test of character, a real challenge.
"I'm delighted with the reaction, delighted we found the answers because earlier in the season we would have folded.
"That's why I'm so pleased because we've learned our lessons from earlier in the campaign."