The change means that the leader of the green party on the council, Duncan Kerr, will now not be mayor in 2020.
A furious row broke out at Monday's meeting when it was expected that councillors would follow tradition and elect current deputy mayor, Councillor John Price as mayor and Councillor Kerr, next in line in seniority, as deputy.
But after confirming Councillor Price as mayor-elect Councillor Peter Cherrington successfully proposed suspending standing orders to elect the deputy, proposing Councillor Mark Jones instead of Councillor Kerr, a move that was voted for.
He said in recent years Oswestry Town Council had become too political.
"Some of the items on the agenda recently have been political and not for the benefit of the people of Oswestry," he said after the meeting.
"Also based on seniority means that people can find themselves becoming mayor, two, three or more times when those who have worked really hard or done a tremendous job will not get to become mayor."
The move drew claims that the move was a political one to stop a member of the Green party becoming mayor, with Councillor Nikki Hughes saying she had never been so angry.
Councillor Cherrington, himself a former mayor, said: "I have always been an independent and this had nothing to do with politics. The town council is, and always should be apolitical."
Those opposing the change included Councillor Olly Rose who said: "With the seniority list we have a fair system where councillors after they have been council members for a period of time have the honour and opportunity to serve as deputy mayor and then mayor.
"All councillors would have to have served long enough to be re-elected at least once before they reach the top of the list.
"By choosing to suspend the standing orders and miss the next person on the list out, certain councillors have taken into their own hands to choose a deputy mayor more in line with their beliefs.
"Acting in this way allows councillors to act on a whim, allow personal or political differences to cloud their judgement rather than sticking to the present system that ensures fairness in allowing all elected members in order of length of service to have the opportunity to represent their town."
Councillor Kerr said after the meeting: "It has been a long standing tradition and indeed the rules of Oswestry town council that the longest serving councillor becomes the deputy mayor and then the mayor, avoiding civic roles from being political appointments.
"Disappointingly this didn't happen and the rules were broken to appoint a Conservative councillor with less service than myself.
"It is a sad day for the town, the council and for Oswestry. At a time when we need to respect differing views the council should not be throwing away a tradition that has served the town well."