Williams was under increasing pressure before his side defeated Italy in the Six Nations last month.
The win over the Azzurri was Williams' first triumph in the tournament and only his third win in charge of the national side.
And McGeechan, who coached the Lions to series victories in Australia and South Africa, is convinced Williams will now be relishing the chance to end Wales' Grand Slam hopes at Murrayfield on Sunday.
The former Scotland and Lions coach said: "The stick Matt Williams has had goes with the territory and it is tough.
"But Matt handled the situation well enough to be looking forward now to the Welsh game."
The win against Italy was far from spectacular but McGeechan insists the two points were more important than anything else.
He added: "Some wins as a coach you call 'uglies' and that's what they are.
"But the most important thing is winning because it has a knock-on effect, to the players' benefit.
"It was especially important for those players who had never won in the Six Nations because they will have a different feel for the tournament now."