With two minutes remaining at the Riverside Stadium, Shots striker Danny Hylton scored his eighth FA Cup goal in four games following Lukas Jutkiewicz's late opener. Hylton still leads the scoring charts but Jutkiewicz ensured he will not add to his tally by sparing Boro a tiring trip to Hampshire with a winner five minutes into stoppage time.
Mowbray said: "Managers only ever want a home tie and in an ideal world it wouldn't be someone from our league. I'd like to play a Premier League side or one of the lower-league teams who've done well. The FA Cup starts getting exciting in the fifth round as you're one game away from the quarter-finals.
He continued: "Let's see who we get. If we play Manchester United or Arsenal the onus is not on us to break them down, it's on them to break us down and if they get frustrated every time a move breaks down we can counter.
"Either is suitable for us. A top team that we can frustrate or a lower league team we can give problems and try to get through to the next round."
Middlesbrough struggled to convert their chances against a resilient Aldershot side, who might have trailed 3-0 at the break if not for the heroics of goalkeeper Jamie Young. And Mowbray has identified midfielder Kieron Dyer, recently released by QPR, as the solution to Boro's lack of creativity.
"I've had some conversations with Kieron," Mowbray added.
"If Kieron was fit and playing for us then he would be a huge asset for our club.
"I've watched some footage of him playing against Gareth Bale for QPR and he did well. He's scored in the FA Cup against West Brom and he's been available, fit and playing. I'll have to see if something can be done and we might be close to that. If Kieron plays football for us and helps win football matches then I think he would be an asset."
Shots boss Dean Holdsworth, meanwhile, was left wondering what might have been after elation turned to disappointment inside 10 frantic minutes. When asked if the result was a cruel one, he said: "It is certainly is for the players and everyone associated with us because I felt we'd earned a replay."