Marco Reus put Dortmund ahead in the 18th minute but the Germany international, criticised for a string of weak performances recently for club and country, saw his second half penalty saved as Dortmund wasted several good chance to score.
The visitors had to wait until Henrikh Mkhitaryan's close-range effort in the 82nd minute to make the points safe for their first win after three draws and a defeat in all competitions.
"You could feel that we had played three draws (including one in the Europa League) and were crushed (5-1) at Bayern Munich and so not everything will work easily," said Tuchel, who had a five-year spell at Mainz with two Europa League qualifications before an acrimonious split in 2014 and a one-year sabbatical.
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"We have the quality and we have the three points. It may have been a bit shaky today but that does not matter. It could not have worked better."
Dortmund's first league win since Sept. 23, took them to 20 points, four behind Bayern who take on Werder Bremen on Saturday.
The 42-year-old Dortmund coach had to endure a nervous first game back at Mainz with his players wasteful once more this season and Mainz hitting the woodwork.
Reus put the visitors ahead with his third goal of the season after a fine combination between Mkhitaryan and Pierre-Emerick Aubameyang, who missed out on setting a league record of scoring in 11 consecutive Bundesliga matches.
Instead of adding another, out-of-form Reus had his penalty saved by Mainz's Loris Karius and the player, along with Aubameyang and Shinji Kagawa, squandered half a dozen more opportunities before Mkhitaryan rounded the keeper to seal their sixth victory in nine matches eight minutes from time.
(Reporting by Karolos Grohmann; editing by Martyn Herman)
