One way could be to add a callback closure to a custom cell and execute it upon cell selection. your view controller would hook this callback up in tableView(_,cellForIndexPath:) or tableView(_, cellWillAppear:)
import UIKit
class SelectionCallbackTableViewCell: UITableViewCell {
    var selectionCallback: (() -> Void)?
    @IBOutlet weak var button: UIButton!
    @IBAction func buttonTapped(sender: UIButton) {
        self.selectionCallback?()
    }
    override func layoutSubviews() {
        super.layoutSubviews()
        self.contentView.addSubview(self.button)
    }
}
register the cell for the tableview. I did it in the storyboard.
import UIKit
class ViewController: UIViewController, UITableViewDataSource, UITableViewDelegate {
    @IBOutlet weak var tableView: UITableView!
    var array = [1,1]
    override func viewDidLoad() {
        super.viewDidLoad()
        for x in stride(from: 2, through: 30, by: 1){
            let i = array[x-2]
            let j = array[x-1]
            array.append(i+j)
        }
    }
    func numberOfSectionsInTableView(tableView: UITableView) -> Int {
        return 1
    }
    func tableView(tableView: UITableView, numberOfRowsInSection section: Int) -> Int {
        return array.count
    }
    func tableView(tableView: UITableView, cellForRowAtIndexPath indexPath: NSIndexPath) -> UITableViewCell {
        let cell = tableView.dequeueReusableCellWithIdentifier("cell", forIndexPath: indexPath) as! SelectionCallbackTableViewCell
        cell.selectionCallback = {
            println("\(self.array[indexPath.row]) at \(indexPath.row) on \(tableView) selected")
        }
        cell.textLabel?.text = "\(self.array[indexPath.row])"
        return cell
    }
}
Now implement the callback closure as needed
cell.selectionCallback = {
    println("\(self.array[indexPath.row]) at \(indexPath.row) on \(tableView) selected")
}
will result in logs like
832040 at 29 on <UITableView: 0x7fe290844200; frame = (0 0; 375 667); clipsToBounds = YES; autoresize = RM+BM; gestureRecognizers = <NSArray: 0x7fe2907878e0>; layer = <CALayer: 0x7fe290711e60>; contentOffset: {0, 697}; contentSize: {375, 1364}> selected
all possible helpful information are present: 
- indexPath
- datasource object
- table view
- cell
if you want delete the cell after successfully informing your server, for a simple solution do:
cell.selectionCallback = {
    println("\(self.array[indexPath.row]) at \(indexPath.row) on \(tableView) selected")
    self.apiManager.deleteObject(obj, success: { response in
        self.array.removeAtIndex(indexPath.row)
        tableView.reloadData()
    }, failure:{ error in 
        // handle error
    })
}
Disclaimer:
Personally I don't recommend to implement datasources as UIViewController. It violates the SOLID principles.  
Instead datasources should reside in their own classes. 
For my own Projects I use an architecture I call "OFAPopulator", that allows me to create independent datasources for each section of a table or collection view. 
I chose to implement it inside the view controller here for brevity. 
I missed the button part. edited.
Example code: https://github.com/vikingosegundo/CellSelectionCallback