Dark-horses Mainz had a penalty saved but were ultimately outgunned by Jürgen Klopp's young team in a tense, sell-out affair at Mainz's Bruchwegstadion.
"We implemented our tactics well against strong opponents and it was a good win for us," beamed Klopp. "However, we should have taken the lead earlier and the penalty save was a key moment in the game. After that, we had luck on our side."
With only a quarter of the season gone, Klopp said thoughts of Dortmund winning a seventh Bundesliga title have not even entered his head.
The win gives Dortmund a one-point lead over Mainz, who drop to second after ten league matches.
With the guests enjoying a few chances in the build up, Dortmund opened the scoring on 26 minutes when 18-year-old Götze pounced on Mainz defender Niko Bungert's mistake and fired home.
Mainz battled to force the equaliser and were awarded a penalty by referee Michael Weiner on 47 minutes when Serbia defender Neven Subotic fouled Hungarian Adam Szalai in the penalty area.
But Mainz midfielder Eugen Polanski had his shot saved by Dortmund captain Roman Weidenfeller and Borussia cleared the danger.
Paraguay striker Barrios scored his fifth league goal of the season when he fired home from a tight angle on 67 minutes and although Mainz fought hard to get back in the game, they struggled against Dortmund's strong defence.
Later Hoffenheim went third in the Bundesliga as they hammered Hannover 96 4-0 with Iceland midfielder Gylfi Sigurdsson scoring twice.
On Saturday, Werder Bremen and Schalke both warmed-up in poor fashion for crunch Champions League clashes on Tuesday as they were both beaten at home.
Werder Bremen are at home to Dutch side FC Twente in Group A, but will face the new Dutch league leaders still reeling from a 3-2 defeat at home to Nuremberg.
Schalke face Hapoel Tel Aviv away in Group B, but they slumped to a 1-0 defeat at home to Leverkusen - the Royal Blues' sixth defeat in 10 games - while Leverkusen are fourth in the table.
Eintracht Frankfurt went fifth after their 3-1 win at ten-man St Pauli.
Wolfsburg striker Edin Dzeko also netted twice in his side's 2-0 win over Stuttgart while Hamburg were upset 3-2 at Cologne as Slovenian striker Milivoje Novakovic scored a hat-trick.
Kaiserslautern beat Mönchengladbach 3-0 while defending German champions Bayern Munich are seventh in the table after a 4-2 win over Freiburg on Friday.