Suzuki had his best game for New York since being obtained in a trade from Seattle last month.
He contributed a two-run single that followed Teixeira's solo homer in the eighth inning -- staking the Yankees to a 6-2 lead -- and added a two-run double in the ninth as New York piled on four more runs.
Nick Swisher and Robinson Cano each supplied two hits and an RBI as the American League East-leading Yankees took the opener of a three-game series.
Freddy Garcia (6-5) earned the victory with six solid innings, yielding two runs on five hits. The right-hander didn't issue a walk and struck out four.
Toronto's Ricky Romero (8-9), meanwhile, lost for the eighth time in nine starts, giving up three runs (two earned) on four hits in a seven-inning effort.
Kelly Johnson went 2-for-4 with a solo homer and two RBI for the Blue Jays.