President Donald Trump's attack on a Syrian air base has signaled a shift in his once-cozy relationship with Russian President Vladimir Putin, former Defense Secretary William Cohen told CNBC on Friday.

Secretary of State Rex Tillerson is scheduled to travel to Moscow on Wednesday, a week after the U.S. military fired 59 Tomahawk missiles on a Syria military air base in response to Tuesday's deadly chemical weapons attack allegedly carried out by Bashar Assad's military on a rebel-held area.

The Syrian strike "sets the tone," Cohen, a Republican who served under former President Bill Clinton, said on "Squawk Box." "Life between us is going to be somewhat different."

After Thursday's U.S. missile strikes, Tillerson said Russia had failed to carry out the 2013 agreement to secure Syrian chemical weapons. He said Moscow was either complicit or incompetent in its ability to uphold that deal.

Russia said the U.S. strikes against Assad's government violated international law. Kremlin spokesman Dmitry Peskov said Putin believed the U.S. attacks on Syria showed aggression against a sovereign state.

Cohen said one strike against Assad's government is not a strategy, but a tactic. He also said Tillerson needs more staff support at the State Department, where many top spots are still unfilled.

"We need to get the State Department really beefed up so that Secretary Tillerson can have his full team to give all the options to the president as he moves forward," Cohen said. "It won't be just military, it has to be economic."