Stead turned a deflected shot home on the break in the 81st minute and Mclean scored six minutes later as Junior Brown’s red card just after the half-hour mark proved costly for the hosts.
Matthew Pennington had earlier given Rovers the lead after just seven minutes but even a win would have seen Tranmere relegated with Notts County drawing 1-1 at Oldham.
Everton loanee Pennington watched his shot go through the hands of Bradford goalkeeper Jon McLaughlin before dropping into the net as Rovers got exactly the start they wanted.
Tempers almost boiled over when the two benches clashed after home striker Ryan Lowe appeared to be pushed inside the Bradford technical area.
And things erupted further when Brown received a straight red card for a 31st-minute challenge on Nathan Doyle.
Stead fired over for Bradford eight minutes into the second half but levelled matters before Mclean sealed the deal after Jean-Louis Akpa Akpro had wasted a chance to win it for Rovers.