The duo engaged in a public spat over the budget and transfer fees spent after Mackay guided Cardiff back into the top flight in May following a 51-year absence.
The drawn out saga finally ended last week when Mackay, who had been at the helm since June 2011, was sacked and replaced on Thursday by former Manchester United striker Ole Gunnar Solskjaer.
"I hope you'll forgive us for that and put some trust in us to build this club to what I think it can be built into," Dalman, who was appointed chairman in July, was quoted as saying by the BBC.
"This club has huge potential but it all has to start at the football pitch. So less noise about non-football and a lot of noise on the football pitch."
Mackay was widely respected for the job he did and hugely popular with fans who have protested in recent weeks against Tan, the equally unpopular Malaysian owner of the club who initially angered supporters by changing their kit from blue shirts to red before the start of last season.
Following Mackay's sacking, Tan said in a statement: "Far too much dirty linen has been exposed to the public gaze but I stress, not by me."
Dalman added he was sorry for the "agony" Cardiff supporters had been put through but that they should now look forward and get behind Solskjaer, who has been given money by Tan to spend on new signings during the January transfer window.
"He knows what he wants and yes we have discussed all that," said Dalman. "The first and only objective is to remain in the Premier League."
Cardiff will face Newcastle United in the third round of the FA Cup on Saturday. In the Premier League, they are hovering above the relegation zone in 17th place with 18 points.
(Writing by Pritha Sarkar, editing by Mitch Phillips)
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