The England midfielder netted the only goal of the game on Wednesday as Arsenal edged out Swansea in a third round replay.
The Gunners will face Championship opposition at the next stage, with a trip to Brighton awaiting Arsene Wenger's troops.
Wilshere is aware of the importance of piecing together a positive cup run, with Arsenal having gone eight years since they last secured major silverware.
He said: "We had a lot of chances in the second half and couldn't put it in. Luckily I was there at the end to do that.
"We know what the FA Cup means to people at the club, it's a great competition and we wanted to be in the next round."
Wilshere is also pleased that his 86th minute effort saw Arsenal avoid the need for extra-time against Swansea, with an important Premier League fixture against capital rivals Chelsea next on their agenda.
He added: "I don't think I'll hit one like that again. It was a great set-up from Olivier Giroud and it was important that we got the goal.
"We didn't want extra-time with the tough game coming up at the weekend."