Yes, but notice WHERE in the constitution that power is granted. It is in Article I and is granted to the Legislative branch. To suspend habeas corpus as Lincoln did was also to usurp a power of the Congress that he did not properly have in his own right.
Since the legislative branch was not in session, the rebellion made it necessary for the Executive branch to act.
When Congress reconvened on July 4th they supported Lincoln's actions.
Going back to Jackson,
Andrew Jackson, who was aleady at odds with Calhoun for other reasons, was enraged. He was said to have told on congressman that South Carolina could adopt resolutions to it hearts content, but 'if one drop of blood be shed there in defiance of the laws of the United States, I will hang the first man of them I can get my hands on to the first tree I can find'. That story may be apocryphal, but he clearly did tell Van Buren that Calhoun should be hanged for a traitor. How serious was he? "I don't believe he would hang anybody do you?" Senator Hayne asked a leading Jacksonian. The reply was: "Well before he invaded Florida on his own hook, few people could have believed that he would hang Arbuthonot and shoot Ambrister (two British subjects)-also on his own authority-could they? I tell you, Hayne, when Jackson begins talking about hanging, they can begin to look for ropes" (Daniel Farber, Lincoln's Constitution, p.61)
Regarding Lincolns actions
And yet, it is difficult to condemn these actions too harshly. Neither later historians nor Lincoln's contemporaries seem to have questioned the urgent need to expand the military. Moreover, Congress did ultimately endorse these actions. On August 5,it almost unanimously passed a bill declaring 'all the acts proclamations and orders of the President' taken after Lincoln took office ' respecting the army and navy of the United States and calling out or relating to the militia or volunteers from the States, are hereby approved and in all respects legalized and made valid (Ibid, p.138)
Regarding Habeas,
Congress ultimately settled the dispute in March 1863 with a statute declaring that the President did have the right to suspend the writ. Under the statue, a military officer could respond to the writ merely by certifying that the person was detained by authority of the president...(Ibid, p.159)